Differential principal factors and Polya property of pure metacyclic fields

TL;DR

This paper generalizes principal factorization theory from pure cubic to pure quintic and metacyclic fields, classifies their types, and investigates the Polya property using Galois cohomology, supported by extensive numerical analysis.

Contribution

It extends Barrucand and Cohn's principal factorization classification to pure quintic and metacyclic fields, establishing new criteria for the Polya property and providing detailed group structure analysis.

Findings

13 possible types of pure metacyclic fields identified

Pure metacyclic fields of only 1 type are not Polya fields

Numerical verification conducted for 900 radicands in range 2 to 999

Abstract

Barrucand and Cohn's theory of principal factorizations in pure cubic fields \(\mathbb{Q}(\sqrt[3]{D})\) and their Galois closures \(\mathbb{Q}(\zeta_3,\sqrt[3]{D})\) with \(3\) types is generalized to pure quintic fields \(L=\mathbb{Q}(\sqrt[5]{D})\) and pure metacyclic fields \(N=\mathbb{Q}(\zeta_5,\sqrt[5]{D})\) with \(13\) possible types. The classification is based on the Galois cohomology of the unit group \(U_N\), viewed as a module over the automorphism group \(\mathrm{Gal}(N/K)\) of \(N\) over the cyclotomic field \(K=\mathbb{Q}(\zeta_5)\), by making use of theorems by Hasse and Iwasawa on the Herbrand quotient of the unit norm index \((U_K:N_{N/K}(U_N))\) by the number \(\#(\mathcal{P}_{N/K}/\mathcal{P}_K)\) of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different \(\mathfrak{D}_{N/K}\). The precise structure of the group of differential principal factors is determined with the aid of kernels of norm homomorphisms and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units \((U_N:U_0)\). Generalizing criteria for the Polya property of Galois closures \(\mathbb{Q}(\zeta_3,\sqrt[3]{D})\) of pure cubic fields \(\mathbb{Q}(\sqrt[3]{D})\) by Leriche and Zantema, we prove that pure metacyclic fields \(N=\mathbb{Q}(\zeta_5,\sqrt[5]{D})\) of only \(1\) type cannot be Polya fields. All theoretical results are underpinned by extensive numerical verifications of the \(13\) possible types and their statistical distribution in the range \(2\le D<10^3\) of \(900\) normalized radicands.

Tables

Table 1. Table 1 . Differential principal factorization types of pure metacyclic fields N 𝑁 N

Type	$U$	$U+1=A+I+R$	$A$	$I$	$R$
${\alpha }_1$	$2$	$3$	$1$	$0$	$2$
${\alpha }_2$	$2$	$3$	$1$	$1$	$1$
${\alpha }_3$	$2$	$3$	$1$	$2$	$0$
${\beta }_1$	$2$	$3$	$2$	$0$	$1$
${\beta }_2$	$2$	$3$	$2$	$1$	$0$
$\gamma$	$2$	$3$	$3$	$0$	$0$
${\delta }_1$	$1$	$2$	$1$	$0$	$1$
${\delta }_2$	$1$	$2$	$1$	$1$	$0$
$\epsilon$	$1$	$2$	$2$	$0$	$0$
${\zeta }_1$	$1$	$2$	$1$	$0$	$1$
${\zeta }_2$	$1$	$2$	$1$	$1$	$0$
$\eta$	$1$	$2$	$2$	$0$	$0$
$\vartheta$	$0$	$1$	$1$	$0$	$0$

Type	$U$	$U+1=A+I+R$	$A$	$I$	$R$
${\alpha }_1$	$2$	$3$	$1$	$0$	$2$
${\alpha }_2$	$2$	$3$	$1$	$1$	$1$
${\alpha }_3$	$2$	$3$	$1$	$2$	$0$
${\beta }_1$	$2$	$3$	$2$	$0$	$1$
${\beta }_2$	$2$	$3$	$2$	$1$	$0$
$\gamma$	$2$	$3$	$3$	$0$	$0$
${\delta }_1$	$1$	$2$	$1$	$0$	$1$
${\delta }_2$	$1$	$2$	$1$	$1$	$0$
$\epsilon$	$1$	$2$	$2$	$0$	$0$
${\zeta }_1$	$1$	$2$	$1$	$0$	$1$
${\zeta }_2$	$1$	$2$	$1$	$1$	$0$
$\eta$	$1$	$2$	$2$	$0$	$0$
$\vartheta$	$0$	$1$	$1$	$0$	$0$

Type	$U$	$U+1=B+T$	$B$	$T$
$\alpha$	$1$	$2$	$1$	$1$
$\beta$	$1$	$2$	$2$	$0$
$\gamma$	$0$	$1$	$1$	$0$

$g$	$1$	$\tau ={\tau }^{-1}$
${\chi }_0\left(g\right)$	$1$	$1$
${\chi }_1\left(g\right)$	$1$	$-1$

$g$	$1$	$\tau ={\tau }^{-3}$	${\tau }^2={\tau }^{-2}$	${\tau }^3={\tau }^{-1}$
${\chi }_0\left(g\right)$	$1$	$1$	$1$	$1$
${\chi }_1\left(g\right)$	$1$	$\sqrt{-1}$	$-1$	$-\sqrt{-1}$
${\chi }_2\left(g\right)$	$1$	$-1$	$1$	$-1$
${\chi }_3\left(g\right)$	$1$	$-\sqrt{-1}$	$-1$	$\sqrt{-1}$

		$N_{N/M}$
	$M$	$\leftarrow$	$N$
$N_{M/K^{+}}$	$\downarrow$	$///$	$\downarrow$	$N_{N/K}$
	$K^{+}$	$\leftarrow$	$K$
		$N_{K/K^{+}}$

Table 7. Table 2. Differential principal factorization types of pure metacyclic fields N 𝑁 N

Type	$U$	$\eta$	$\zeta$	$A$	$I$	$R$
${\alpha }_1$	$2$	$-$	$-$	$1$	$0$	$2$
${\alpha }_2$	$2$	$-$	$-$	$1$	$1$	$1$
${\alpha }_3$	$2$	$-$	$-$	$1$	$2$	$0$
${\beta }_1$	$2$	$-$	$-$	$2$	$0$	$1$
${\beta }_2$	$2$	$-$	$-$	$2$	$1$	$0$
$\gamma$	$2$	$-$	$-$	$3$	$0$	$0$
${\delta }_1$	$1$	$\times$	$-$	$1$	$0$	$1$
${\delta }_2$	$1$	$\times$	$-$	$1$	$1$	$0$
$\epsilon$	$1$	$\times$	$-$	$2$	$0$	$0$
${\zeta }_1$	$1$	$-$	$\times$	$1$	$0$	$1$
${\zeta }_2$	$1$	$-$	$\times$	$1$	$1$	$0$
$\eta$	$1$	$-$	$\times$	$2$	$0$	$0$
$\vartheta$	$0$	$\times$	$\times$	$1$	$0$	$0$

Table 8. Table 3. Absolute frequencies of differential principal factorization types

Type	$100$	$200$	$300$	$400$	$500$	$600$	$700$	$800$	$900$	$1000$	$\%$
${\alpha }_1$	$1$	$2$	$3$	$4$	$5$	$5$	$5$	$9$	$9$	$9$
${\alpha }_2$	$10$	$17$	$23$	$30$	$35$	$42$	$52$	$57$	$63$	$75$	$8.3$
${\alpha }_3$	$0$	$0$	$0$	$1$	$1$	$3$	$5$	$5$	$7$	$8$
${\beta }_1$	$0$	$2$	$4$	$7$	$8$	$11$	$15$	$18$	$22$	$23$
${\beta }_2$	$7$	$24$	$40$	$54$	$80$	$94$	$108$	$126$	$146$	$161$	$17.9$
$\gamma$	$25$	$55$	$88$	$117$	$148$	$187$	$222$	$259$	$290$	$324$	$36.0$
${\delta }_1$	$0$	$0$	$1$	$1$	$3$	$4$	$4$	$4$	$6$	$7$
${\delta }_2$	$8$	$14$	$19$	$23$	$31$	$35$	$38$	$44$	$51$	$53$	$5.9$
$\epsilon$	$26$	$45$	$67$	$95$	$110$	$128$	$150$	$165$	$184$	$208$	$23.1$
${\zeta }_1$	$0$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
${\zeta }_2$	$0$	$0$	$0$	$0$	$0$	$1$	$1$	$4$	$4$	$5$
$\eta$	$1$	$2$	$4$	$5$	$5$	$6$	$6$	$6$	$6$	$7$
$\vartheta$	$3$	$6$	$8$	$9$	$11$	$13$	$15$	$17$	$18$	$19$
Total	$81$	$168$	$258$	$347$	$438$	$530$	$622$	$715$	$807$	$900$	$100.0$

Table 9. Table 4. 38 38 38 pure metacyclic fields with normalized radicands 0 < D < 50 0 𝐷 50 0<D<50

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
1	*$2$		1b	$5^2{2}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
2	$3$		1b	$5^2{3}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
3	*$5$		1a	$5^6$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
4	*$6$	$2\cdot 3$	1b	$5^2{2}^4{3}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
5	*$7$		2	$7^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
6	*$10$	$2\cdot 5$	1a	$5^6{2}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
7	*$11$		1b	$5^2{11}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_1$
8	$12$	$2^2\cdot 3$	1b	$5^2{2}^4{3}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
9	$13$		1b	$5^2{13}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
10	*$14$	$2\cdot 7$	1b	$5^2{2}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
11	$15$	$3\cdot 5$	1a	$5^6{3}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
12	$17$		1b	$5^2{17}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
13	*$18$	$2\cdot 3^2$	2	$2^4{3}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
14	*$19$		1b	$5^2{19}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
15	$20$	$2^2\cdot 5$	1a	$5^6{2}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
16	$21$	$3\cdot 7$	1b	$5^2{3}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
17	*$22$	$2\cdot 11$	1b	$5^2{2}^4{11}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot 5,𝒦$
18	$23$		1b	$5^2{23}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
19	$26$	$2\cdot 13$	2	$2^4{13}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
20	$28$	$2^2\cdot 7$	1b	$5^2{2}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
21	$29$		1b	$5^2{29}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
22	*$30$	$2\cdot 3\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
23	*$31$		1b	$5^2{31}^4$	$1$	$2$	$3$	$5$	$2$	$(-,-,\otimes ,-)$	${\alpha }_1$	${𝔎}_1,{𝔎}_2$
24	*$33$	$3\cdot 11$	1b	$5^2{3}^4{11}^4$	$3$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
25	$34$	$2\cdot 17$	1b	$5^2{2}^4{17}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
26	*$35$	$5\cdot 7$	1a	$5^6{7}^4$	$4$	$0$	$0$	$1$	$6$	$(-,-,-,\otimes )$	$\eta$
27	$37$		1b	$5^2{37}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
28	*$38$	$2\cdot 19$	1b	$5^2{2}^4{19}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,𝒦$
29	$39$	$3\cdot 13$	1b	$5^2{3}^4{13}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
30	$40$	$2^3\cdot 5$	1a	$5^6{2}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
31	$41$		1b	$5^2{41}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_2$
32	*$42$	$2\cdot 3\cdot 7$	1b	$5^2{2}^4{3}^4{7}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 5,3\cdot 5^2$
33	$43$		2	${43}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
34	$44$	$2^2\cdot 11$	1b	$5^2{2}^4{11}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$2\cdot 5,𝒦$
35	$45$	$3^2\cdot 5$	1a	$5^6{3}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
36	$46$	$2\cdot 23$	1b	$5^2{2}^4{23}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
37	$47$		1b	$5^2{47}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
38	$48$	$2^4\cdot 3$	1b	$5^2{2}^4{3}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$

Table 10. Table 5. 43 43 43 pure metacyclic fields with normalized radicands 50 < D < 100 50 𝐷 100 50<D<100

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
39	$51$	$3\cdot 17$	2	$3^4{17}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
40	$52$	$2^2\cdot 13$	1b	$5^2{2}^4{13}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
41	$53$		1b	$5^2{53}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
42	*$55$	$5\cdot 11$	1a	$5^6{11}^4$	$4$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
43	$56$	$2^3\cdot 7$	1b	$5^2{2}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
44	*$57$	$3\cdot 19$	2	$3^4{19}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
45	$58$	$2\cdot 29$	1b	$5^2{2}^4{29}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$29\cdot 5^2,𝒦$
46	$59$		1b	$5^2{59}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
47	$60$	$2^2\cdot 3\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
48	$61$		1b	$5^2{61}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_2$
49	$62$	$2\cdot 31$	1b	$5^2{2}^4{31}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$5,𝒦$
50	$63$	$3^2\cdot 7$	1b	$5^2{3}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
51	$65$	$5\cdot 13$	1a	$5^6{13}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
52	*$66$	$2\cdot 3\cdot 11$	1b	$5^2{2}^4{3}^4{11}^4$	$13$	$1$	$2$	$5$	$6$	$(-,-,\times ,-)$	$\gamma$	$2\cdot 5,3\cdot 5^3$
53	$67$		1b	$5^2{67}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
54	$68$	$2^2\cdot 17$	2	$2^4{17}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
55	$69$	$3\cdot 23$	1b	$5^2{3}^4{23}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
56	*$70$	$2\cdot 5\cdot 7$	1a	$5^6{2}^4{7}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
57	$71$		1b	$5^2{71}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_1$
58	$73$		1b	$5^2{73}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
59	$74$	$2\cdot 37$	2	$2^4{37}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
60	$75$	$3\cdot 5^2$	1a	$5^6{3}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
61	$76$	$2^2\cdot 19$	2	$2^4{19}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
62	*$77$	$7\cdot 11$	1b	$5^2{7}^4{11}^4$	$4$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11\cdot 5^3,𝒦$
63	*$78$	$2\cdot 3\cdot 13$	1b	$5^2{2}^4{3}^4{13}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$3,2\cdot 5^3$
64	$79$		1b	$5^2{79}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
65	$80$	$2^4\cdot 5$	1a	$5^6{2}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
66	*$82$	$2\cdot 41$	2	$2^4{41}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
67	$83$		1b	$5^2{83}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
68	$84$	$2^2\cdot 3\cdot 7$	1b	$5^2{2}^4{3}^4{7}^4$	$12$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 5,3\cdot 5$
69	$85$	$5\cdot 17$	1a	$5^6{17}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
70	$86$	$2\cdot 43$	1b	$5^2{2}^4{43}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
71	$87$	$3\cdot 29$	1b	$5^2{3}^4{29}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$29,ℒ$
72	$88$	$2^3\cdot 11$	1b	$5^2{2}^4{11}^4$	$3$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
73	$89$		1b	$5^2{89}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
74	$90$	$2\cdot 3^2\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
75	$91$	$7\cdot 13$	1b	$5^2{7}^4{13}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
76	$92$	$2^2\cdot 23$	1b	$5^2{2}^4{23}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
77	$93$	$3\cdot 31$	2	$3^4{31}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
78	$94$	$2\cdot 47$	1b	$5^2{2}^4{47}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
79	*$95$	$5\cdot 19$	1a	$5^6{19}^4$	$4$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
80	$97$		1b	$5^2{97}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
81	$99$	$3^2\cdot 11$	2	$3^4{11}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$

Table 11. Table 6. 44 44 44 pure metacyclic fields with normalized radicands 100 < D ≤ 150 100 𝐷 150 100<D\leq 150

No.	$D$	Factors	S	$f^4$	$m$	$V_L$	$V_M$	$V_N$	$E$	$(1,2,4,5)$	T	P
82	*$101$		2	${101}^4$	$1$	$1$	$2$	$4$	$5$	$(-,-,\otimes ,\otimes )$	${\zeta }_1$	${𝔎}_2$
83	$102$	$2\cdot 3\cdot 17$	1b	$5^2{2}^4{3}^4{17}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2,3\cdot 5$
84	$103$		1b	$5^2{103}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
85	$104$	$2^3\cdot 13$	1b	$5^2{2}^4{13}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
86	$105$	$3\cdot 5\cdot 7$	1a	$5^6{3}^4{7}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
87	$106$	$2\cdot 53$	1b	$5^2{2}^4{53}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
88	$107$		2	${107}^4$	$1$	$0$	$0$	$0$	$5$	$(-,-,-,\otimes )$	$\vartheta$
89	$109$		1b	$5^2{109}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$ℒ$
90	*$110$	$2\cdot 5\cdot 11$	1a	$5^6{2}^4{11}^4$	$16$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11,ℒ$
91	$111$	$3\cdot 37$	1b	$5^2{3}^4{37}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
92	$112$	$2^4\cdot 7$	1b	$5^2{2}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
93	$113$		1b	$5^2{113}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
94	*$114$	$2\cdot 3\cdot 19$	1b	$5^2{2}^4{3}^4{19}^4$	$13$	$1$	$2$	$5$	$6$	$(-,\times ,-,-)$	$\gamma$	$2\cdot 5^3,3\cdot 5^3$
95	$115$	$5\cdot 23$	1a	$5^6{23}^4$	$4$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
96	$116$	$2^2\cdot 29$	1b	$5^2{2}^4{29}^4$	$3$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$29\cdot 5,𝒦$
97	$117$	$3^2\cdot 13$	1b	$5^2{3}^4{13}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
98	$118$	$2\cdot 59$	2	$2^4{59}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
99	$119$	$7\cdot 17$	1b	$5^2{7}^4{17}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
100	$120$	$2^3\cdot 3\cdot 5$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
101	$122$	$2\cdot 61$	1b	$5^2{2}^4{61}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$61\cdot 5^3,𝒦$
102	*$123$	$3\cdot 41$	1b	$5^2{3}^4{41}^4$	$3$	$2$	$3$	$6$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_2$
103	$124$	$2^2\cdot 31$	2	$2^4{31}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
104	*$126$	$2\cdot 3^2\cdot 7$	2	$2^4{3}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
105	$127$		1b	$5^2{127}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
106	$129$	$3\cdot 43$	1b	$5^2{3}^4{43}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
107	$130$	$2\cdot 5\cdot 13$	1a	$5^6{2}^4{13}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
108	*$131$		1b	$5^2{131}^4$	$1$	$2$	$2$	$4$	$1$	$(-,-,\otimes ,-)$	${\alpha }_2$	$ℒ,{𝔎}_2$
109	*$132$	$2^2\cdot 3\cdot 11$	2	$2^4{3}^4{11}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$11,ℒ$
110	*$133$	$7\cdot 19$	1b	$5^2{7}^4{19}^4$	$4$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$7,ℒ$
111	$134$	$2\cdot 67$	1b	$5^2{2}^4{67}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
112	$136$	$2^3\cdot 17$	1b	$5^2{2}^4{17}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
113	$137$		1b	$5^2{137}^4$	$1$	$0$	$0$	$0$	$5$	$(\times ,-,-,-)$	$\epsilon$
114	$138$	$2\cdot 3\cdot 23$	1b	$5^2{2}^4{3}^4{23}^4$	$13$	$1$	$2$	$5$	$6$	$(\times ,-,-,-)$	$\gamma$	$2\cdot 5^2,3$
115	*$139$		1b	$5^2{139}^4$	$1$	$1$	$1$	$3$	$4$	$(-,\otimes ,-,-)$	${\beta }_2$	$5,ℒ$
116	*$140$	$2^2\cdot 5\cdot 7$	1a	$5^6{2}^4{7}^4$	$16$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$7$
117	*$141$	$3\cdot 47$	1b	$5^2{3}^4{47}^4$	$3$	$1$	$2$	$4$	$5$	$(\times ,-,-,-)$	$\epsilon$	$47\cdot 5$
118	$142$	$2\cdot 71$	1b	$5^2{2}^4{71}^4$	$3$	$1$	$1$	$3$	$4$	$(-,-,(\times ),-)$	${\beta }_2$	$71\cdot 5^2,𝒦$
119	$143$	$11\cdot 13$	2	${11}^4{13}^4$	$1$	$1$	$1$	$2$	$3$	$(-,-,\otimes ,-)$	${\alpha }_2$	$𝒦,{𝔎}_1$
120	$145$	$5\cdot 29$	1a	$5^6{29}^4$	$4$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,-)$	${\delta }_2$	$𝒦$
121	$146$	$2\cdot 73$	1b	$5^2{2}^4{73}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
122	$147$	$3\cdot 7^2$	1b	$5^2{3}^4{7}^4$	$4$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
123	$148$	$2^2\cdot 37$	1b	$5^2{2}^4{37}^4$	$3$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$
124	*$149$		2	${149}^4$	$1$	$1$	$1$	$2$	$3$	$(-,\otimes ,-,\times )$	${\delta }_2$	$ℒ$
125	$150$	$2\cdot 3\cdot 5^2$	1a	$5^6{2}^4{3}^4$	$16$	$0$	$0$	$1$	$6$	$(\times ,-,-,-)$	$\gamma$

Equations

$${\mathfrak{b}}_{p^f}(F):=\prod \{\mathfrak{p}\in {\mathbb{P}}_F\mid N_{F/\mathbb{Q}}(\mathfrak{p})=p^f\}.$$

$$(\forall p\in \mathbb{P})(\forall f\in \mathbb{N})(\exists \Gamma \in F){\mathfrak{b}}_{p^f}(F)=\Gamma {\mathcal{O}}_F.$$

$$\#(H^0(G,U_N))=(U_K:N_{N/K}(U_N))=5^U\text{ with }0\le U\le 2$$

$$H^1(G,U_N)$$

$$\simeq ({\mathcal{P}}_{L/\mathbb{Q}}/{\mathcal{P}}_{\mathbb{Q}})\times \left(({\mathcal{P}}_{M/K^{+}}/{\mathcal{P}}_{K^{+}})\cap \ker (N_{M/L})\right)\times \left(({\mathcal{P}}_{N/K}/{\mathcal{P}}_K)\cap \ker (N_{N/M})\right)$$

$$(\forall p\in \mathbb{P},p\text{ ramified in }N/K)(\exists \alpha \in L)N_{L/\mathbb{Q}}(\alpha )=p.$$

$${\mathfrak{b}}_q(L){\mathcal{O}}_N=\mathcal{Q}{\mathcal{O}}_N=\mathfrak{Q}={\mathfrak{b}}_{q^4}(N).$$

$${\mathfrak{b}}_q(L)=x{\mathcal{O}}_L⟹{\mathfrak{b}}_{q^4}(N)=x{\mathcal{O}}_N,$$

$${\mathfrak{b}}_{q^4}(N)=\Gamma {\mathcal{O}}_N⟹N_{N/L}(\Gamma ){\mathcal{O}}_L=N_{N/L}(\mathfrak{Q})={\mathcal{Q}}^4={\mathfrak{b}}_q(L{)}^4.$$

$${\mathfrak{b}}_{\ell }(L){\mathcal{O}}_N=\mathcal{L}{\mathcal{O}}_N={\mathfrak{L}}_1{\mathfrak{L}}_2={\mathfrak{b}}_{{\ell }^2}(N).$$

$${\mathfrak{b}}_{\ell }(L)=x{\mathcal{O}}_L⟹{\mathfrak{b}}_{{\ell }^2}(N)=x{\mathcal{O}}_N,$$

$${\mathfrak{b}}_{{\ell }^2}(N)=\Gamma {\mathcal{O}}_N⟹N_{N/L}(\Gamma ){\mathcal{O}}_L=N_{N/L}({\mathfrak{L}}_1{\mathfrak{L}}_2)={\mathcal{L}}^4={\mathfrak{b}}_{\ell }(L{)}^4.$$

$${\mathfrak{b}}_{\ell }(L){\mathcal{O}}_N=\mathcal{L}{\mathcal{O}}_N={\mathfrak{L}}_1\cdots {\mathfrak{L}}_4={\mathfrak{b}}_{\ell }(N).$$

$${\mathfrak{b}}_{\ell }(L)=x{\mathcal{O}}_L⟹{\mathfrak{b}}_{\ell }(N)=x{\mathcal{O}}_N,$$

$${\mathfrak{b}}_{\ell }(N)=\Gamma {\mathcal{O}}_N⟹N_{N/L}(\Gamma ){\mathcal{O}}_L=N_{N/L}({\mathfrak{L}}_1\cdots {\mathfrak{L}}_4)={\mathcal{L}}^4={\mathfrak{b}}_{\ell }(L{)}^4.$$

$$\xi {\mathcal{O}}_N={\mathfrak{b}}_5(K){\mathcal{O}}_N=\mathfrak{p}{\mathcal{O}}_N={\mathfrak{P}}^5={\mathfrak{b}}_5(N{)}^5,$$

$${\mathfrak{b}}_5(L){\mathcal{O}}_N=\mathcal{P}{\mathcal{O}}_N={\mathfrak{P}}^4={\mathfrak{b}}_5(N{)}^4.$$

$${\mathfrak{b}}_5(L)=x{\mathcal{O}}_L⟹{\mathfrak{b}}_5(N{)}^4=x{\mathcal{O}}_N⟹{\mathfrak{b}}_5(N)\text{ is principal},$$

$${\mathfrak{b}}_5(N{)}^4=\Gamma {\mathcal{O}}_N⟹N_{N/L}(\Gamma ){\mathcal{O}}_L=N_{N/L}({\mathfrak{P}}^4)={\mathcal{P}}^4={\mathfrak{b}}_5(L{)}^4,$$

$$(\forall p\in \mathbb{P},p\text{ ramified in }N/\mathbb{Q}({\zeta }_3))(\exists \alpha \in L)N_{L/\mathbb{Q}}(\alpha )=p.$$

$$\mathrm{Gal}(N/\mathbb{Q})=⟨\sigma ,\tau \mid {\sigma }^5=1,{\tau }^4=1,\sigma \tau =\tau {\sigma }^2⟩\simeq M_5=C_5⋊C_4$$

$$L_j=L^{{\sigma }^j}=\mathrm{Fix}(⟨{\sigma }^{-j}\tau {\sigma }^j⟩),M_j=M^{{\sigma }^j}=\mathrm{Fix}(⟨{\sigma }^{-j}{\tau }^2{\sigma }^j⟩),\text{ with }0\le j\le 4.$$

$$D^{(1)}$$

$$D^{(2)}$$

$$D^{(3)}$$

$$D^{(4)}$$

$$D^{(k)}\equiv D^k({\mathrm{mod}}^{\times }{\mathbb{Z}}^p)\text{ for each }1\le k\le p-1.$$

$$L=\mathbb{Q}(\text{root}p\text{of}D)\simeq \mathbb{Q}(\text{root}p\text{of}{D}^{(k)})\text{ for all }1\le k\le p-1.\text{qed}$$

$$f^4=\{\begin{array}{ll}{5}^2{R}^4& \text{ if }D\not\equiv \pm 1,\pm 7(\mathrm{mod}{5}^2)\text{ (field of the first species),}\\ R^4& \text{ if }D\equiv \pm 1,\pm 7(\mathrm{mod}{5}^2)\text{ (field of the second species).}\end{array}$$

$$d_N=d_K^5\cdot f^{16}=\{\begin{array}{ll}=5^{23}{R}^{16}& \text{ for a field of the first species,}\\ =5^{15}{R}^{16}& \text{ for a field of the second species,}\end{array}$$

$$d_M=5d_K^2\cdot f^8=\{\begin{array}{ll}=5^{11}{R}^8& \text{ for a field of the first species,}\\ =5^7{R}^8& \text{ for a field of the second species,}\end{array}$$

$$d_L=d_K\cdot f^4=\{\begin{array}{ll}=5^5{R}^4& \text{ for a field of the first species,}\\ =5^3{R}^4& \text{ for a field of the second species.}\end{array}$$

Full text

Research supported by the Austrian Science Fund (FWF): projects J 0497-PHY and P 26008-N25

1. Introduction and Main Theorems

Let $F$ be an algebraic number field. For each prime number $p\in\mathbb{P}$ and each integer exponent $f\geq 1$, let the Ostrowski ideal [25] of $F$ for the prime power $p^{f}$ be defined as the product of all prime ideals of $F$ with norm $p^{f}$, that is

[TABLE]

According to Zantema [37], $F$ is called a Polya field [30], if the Ostrowski ideals of $F$ for arbitrary prime powers are principal, that is

[TABLE]

The aim of the present paper is to provide a necessary and sufficient condition for the Polya property and a classification of all pure metacyclic fields $N=\mathbb{Q}(\zeta_{5},\sqrt[5]{D})$, with $\zeta_{5}=\exp(2\pi\sqrt{-1}/5)$ and $D\geq 2$ a fifth power free integer, into precisely $13$ exhaustive and mutually exclusive types, according to the Galois cohomology of the unit group $U_{N}$, viewed as a module over the automorphism group $G=\mathrm{Gal}(N/K)$ of $N$ over the cyclotomic field $K=\mathbb{Q}(\zeta_{5})$. Each type is characterized uniquely by the [math]-th cohomology, i.e. the unit norm index,

[TABLE]

as the primary classification invariant, and by the triplet $(A,I,R)$ of $\mathbb{F}_{5}$-dimensions of the $5$-elementary abelian components $(\mathcal{P}_{L/\mathbb{Q}}:\mathcal{P}_{\mathbb{Q}})=5^{A}$, $\#\left((\mathcal{P}_{M/K^{+}}/\mathcal{P}_{K^{+}})\cap\ker(N_{M/L})\right)=5^{I}$, and $\#\left((\mathcal{P}_{N/K}/\mathcal{P}_{K})\cap\ker(N_{N/M})\right)=5^{R}$ in the direct product decomposition

[TABLE]

of the $1$-st cohomology, i.e. the group of primitive ambiguous principal ideals of $N/K$, as the secondary classification invariant, where $L=\mathbb{Q}(\sqrt[5]{D})$ denotes the pure quintic subfield of $N$ and $M=\mathbb{Q}(\sqrt{5},\sqrt[5]{D})$ denotes the maximal real subfield of $N$, which contains $K^{+}=\mathbb{Q}(\sqrt{5})$.

Since the Polya property can be proved independently of the classification, we immediately state and prove our first Main Theorem concerning Polya fields, reproved later in Theorem 10.5.

Theorem 1.1**.**

A pure metacyclic field $N=\mathbb{Q}(\zeta_{5},\sqrt[5]{D})$ of absolute degree $[N:\mathbb{Q}]=20$ with $5$-th power free radicand $D\in\mathbb{Z}$, $D\geq 2$, is a Polya field if and only if

[TABLE]

Remark 1.1**.**

It will turn out that only pure metacyclic fields $N$ having the single differential principal factorization type $\alpha_{3}$ in Theorem 1.3 cannot possess the Polya property. For fields $N$ of the other $12$ types we shall give conditions in terms of the prime factorization of the class field theoretic conductor of the Kummer extension $N/K$ which are necessary and sufficient for the Polya property of $N$.

Proof.

Since the absolute extension $N/\mathbb{Q}$ is Galois, it suffices to consider Ostrowski ideals of $N$ for powers of primes which ramify in $N/\mathbb{Q}$ [37]. We reduce the principal ideal conditions for Ostrowski ideals of the normal field $N$ to principal ideal conditions for Ostrowski ideals of the non-Galois field $L$. Then we use the equivalence $\mathfrak{b}_{p}(L)=\alpha\mathcal{O}_{L}$ $\Longleftrightarrow$ $N_{L/\mathbb{Q}}(\alpha)=p$ for $\alpha\in L$ and $p\in\mathbb{P}$ ramified in $L/\mathbb{Q}$.

Firstly, if $q\equiv\pm 2\,(\mathrm{mod}\,5)$, then $q\mathcal{O}_{L}=\mathcal{Q}^{5}$, $N(\mathcal{Q})=q$, $\mathfrak{b}_{q}(L)=\mathcal{Q}$, and $q\mathcal{O}_{N}=\mathfrak{Q}^{5}$, $N(\mathfrak{Q})=q^{4}$, $\mathfrak{b}_{q^{4}}(N)=\mathfrak{Q}$, and

[TABLE]

Thus, on the one hand, for $x\in L$,

[TABLE]

and on the other hand, for $\Gamma\in N$,

[TABLE]

If $\mathcal{Q}$ were not principal, then the class $\mathcal{Q}\mathcal{P}_{L}\in\mathrm{Cl}(L)$ would have order $2$ or $4$ in $\mathrm{Cl}_{2}(L)$ and would capitulate in $N$, $T_{N/L}(\mathcal{Q}\mathcal{P}_{L})=(\mathcal{Q}\mathcal{O}_{N})\mathcal{P}_{N}=\mathfrak{Q}\mathcal{P}_{N}=1$, which is impossible [34].

Secondly, if $\ell\equiv-1\,(\mathrm{mod}\,5)$, then $\ell\mathcal{O}_{L}=\mathcal{L}^{5}$, $N(\mathcal{L})=\ell$, $\mathfrak{b}_{\ell}(L)=\mathcal{L}$, and $\ell\mathcal{O}_{N}=\mathfrak{L}_{1}^{5}\mathfrak{L}_{2}^{5}$, $N(\mathfrak{L}_{i})=\ell^{2}$, $\mathfrak{b}_{\ell^{2}}(N)=\mathfrak{L}_{1}\mathfrak{L}_{2}$, and

[TABLE]

Thus, on the one hand, for $x\in L$,

[TABLE]

and on the other hand, for $\Gamma\in N$,

[TABLE]

If $\mathcal{L}$ were not principal, then the class $\mathcal{L}\mathcal{P}_{L}\in\mathrm{Cl}(L)$ would have order $2$ or $4$ in $\mathrm{Cl}_{2}(L)$ and would capitulate in $N$, $T_{N/L}(\mathcal{L}\mathcal{P}_{L})=(\mathcal{L}\mathcal{O}_{N})\mathcal{P}_{N}=(\mathfrak{L}_{1}\mathfrak{L}_{2})\mathcal{P}_{N}=1$, which is impossible [34].

Next, if $\ell\equiv+1\,(\mathrm{mod}\,5)$, then $\ell\mathcal{O}_{L}=\mathcal{L}^{5}$, $N(\mathcal{L})=\ell$, $\mathfrak{b}_{\ell}(L)=\mathcal{L}$, and $\ell\mathcal{O}_{N}=\mathfrak{L}_{1}^{5}\cdots\mathfrak{L}_{4}^{5}$, $N(\mathfrak{L}_{i})=\ell$, $\mathfrak{b}_{\ell}(N)=\mathfrak{L}_{1}\cdots\mathfrak{L}_{4}$, and

[TABLE]

Thus, on the one hand, for $x\in L$,

[TABLE]

and on the other hand, for $\Gamma\in N$,

[TABLE]

If $\mathcal{L}$ were not principal, then the class $\mathcal{L}\mathcal{P}_{L}\in\mathrm{Cl}(L)$ would have order $2$ or $4$ in $\mathrm{Cl}_{2}(L)$ and would capitulate in $N$, $T_{N/L}(\mathcal{L}\mathcal{P}_{L})=(\mathcal{L}\mathcal{O}_{N})\mathcal{P}_{N}=(\mathfrak{L}_{1}\cdots\mathfrak{L}_{4})\mathcal{P}_{N}=1$, which is impossible [34].

Finally, the investigation of the special prime $5$ must be divided in two parts. Generally, $5\mathcal{O}_{K}=\mathfrak{p}^{4}$, $N(\mathfrak{p})=5$, $\mathfrak{b}_{5}(K)=\mathfrak{p}$, and, since $K$ is a Polya field [37], $\mathfrak{b}_{5}(K)=\xi\mathcal{O}_{K}$ for $\xi\in K$.

If $5$ divides the fourth power of the conductor of $N/K$, then $5\mathcal{O}_{L}=\mathcal{P}^{5}$, $N(\mathcal{P})=5$, $\mathfrak{b}_{5}(L)=\mathcal{P}$, and $5\mathcal{O}_{N}=\mathfrak{P}^{20}$, $N(\mathfrak{P})=5$, $\mathfrak{b}_{5}(N)=\mathfrak{P}$, and

[TABLE]

whence $\mathfrak{b}_{5}(N)$ is principal $\Longleftrightarrow$ $\mathfrak{b}_{5}(N)^{4}$ is principal in $N$. Furthermore,

[TABLE]

Thus on the one hand, for $x\in L$,

[TABLE]

and on the other hand, for $\Gamma\in N$,

[TABLE]

and thus $\mathfrak{b}_{5}(L)=\mathcal{P}=\mathcal{P}^{5}/\mathcal{P}^{4}=5\mathcal{O}_{L}/N_{N/L}(\Gamma)\mathcal{O}_{L}=(5/N_{N/L}(\Gamma))\mathcal{O}_{L}$ is principal.

If $5$ does not divide the fourth power of the conductor of $N/K$, then $5\mathcal{O}_{L}=\mathcal{P}_{1}\mathcal{P}_{2}^{4}$, $N(\mathcal{P}_{i})=5$, $\mathfrak{b}_{5}(L)=\mathcal{P}_{1}\mathcal{P}_{2}$, and $5\mathcal{O}_{N}=\mathfrak{P}_{1}^{4}\cdots\mathfrak{P}_{5}^{4}$, $N(\mathfrak{P}_{i})=5$, $\mathfrak{b}_{5}(N)=\mathfrak{P}_{1}\cdots\mathfrak{P}_{5}$. Since $\mathfrak{p}^{4}\mathcal{O}_{N}=5\mathcal{O}_{N}=\mathfrak{P}_{1}^{4}\cdots\mathfrak{P}_{5}^{4}$, we see that $\mathfrak{b}_{5}(N)=\mathfrak{P}_{1}\cdots\mathfrak{P}_{5}=\mathfrak{p}\mathcal{O}_{N}=\mathfrak{b}_{5}(K)\mathcal{O}_{N}=\xi\mathcal{O}_{N}$ is certainly principal. ∎

Leriche [15, 16] has proved the following analogue of our Theorem 1.1, giving a necessary and sufficient criterion for the Polya property of Galois closures $\mathbb{Q}(\zeta_{3},\sqrt[3]{D})$ of pure cubic fields $\mathbb{Q}(\sqrt[3]{D})$.

Theorem 1.2**.**

The normal field $N=\mathbb{Q}(\zeta_{3},\sqrt[3]{D})$ of a pure cubic field $L=\mathbb{Q}(\sqrt[3]{D})$ with cube free radicand $D\in\mathbb{Z}$, $D\geq 2$, is a Polya field if and only if

[TABLE]

Now we state our second Main Theorem concerning the classification.

Theorem 1.3**.**

Each pure metacyclic field $N=\mathbb{Q}(\zeta_{5},\sqrt[5]{D})$ of absolute degree $[N:\mathbb{Q}]=20$ with $5$-th power free radicand $D\in\mathbb{Z}$, $D\geq 2$, belongs to precisely one of the $13$ differential principal factorization types in Table 1, in dependence on the invariant $U$ and the triplet $(A,I,R)$.

The types $\delta_{1}$, $\delta_{2}$, $\varepsilon$ are characterized additionally by $\zeta_{5}\not\in N_{N/K}(U_{N})$, and the types $\zeta_{1}$, $\zeta_{2}$, $\eta$ by $\zeta_{5}\in N_{N/K}(U_{N})$.

Proof.

The proof will be developed in a sequence of partial results, as explained in the following summary of the layout of this paper. ∎

In § 2, basic arithmetic and Galois theory of pure quintic fields and their pure metacyclic normal closures are recalled, normalized radicands are introduced, and formulas for conductors and discriminants are given. In §§ 3 and 4, differential principal factorizations are explained in a series of steps. Starting with a general definition of ambiguous ideals in arbitrary (not necessarily normal) extensions, we define differential factors as primitive ambiguous ideals by comparing their structure with the shape of the relative different. Further, we investigate kernels of ideal norm homomorphisms and $\mathbb{F}_{5}$-dimensions. In §§ 6 and 7, Galois cohomology is used to prove the Main Classification Theorem 6.1 and $\tau$-invariance is analyzed. Finally, § 10 establishes the Main Theorem 10.5 and underpins all theoretical results in the previous sections by concrete numerical examples for each of the $13$ differential principal factorization types and the statistical distribution of their occurrence in the range $2\leq D<10^{3}$ of $900$ normalized radicands.

2. Pure metacyclic fields

Let $q_{1},\ldots,q_{s}$ be pairwise distinct primes such that $s\geq 1$ and $5$ may be among them. Denote by $L=\mathbb{Q}(\sqrt[5]{D})$ the pure quintic number field with fifth power free radicand $D=q_{1}^{e_{1}}\cdots q_{s}^{e_{s}}$, where the exponents are integers $1\leq e_{i}\leq 4$. The field $L$ is generated by adjoining the unique real and irrational solution of the pure quintic equation $X^{5}-D=0$ to the rational number field $\mathbb{Q}$. It is a non-Galois algebraic number field with signature $(1,2)$ and thus possesses four conjugate and isomorphic complex fields $L_{j}=\mathbb{Q}(\zeta^{j}\cdot\sqrt[5]{D})$, $1\leq j\leq 4$, where $\zeta=\zeta_{5}$ denotes a primitive fifth root of unity, for instance $\zeta=\exp(2\pi\sqrt{-1}/5)$. All arithmetical invariants of the $L_{j}$, $1\leq j\leq 4$, coincide with those of $L$.

The normal closure of $L$ is the compositum $N=L\cdot K=\mathbb{Q}(\sqrt[5]{D},\zeta)$ of $L=\mathbb{Q}(\sqrt[5]{D})$ with the cyclotomic field $K=\mathbb{Q}(\zeta)$. $N$ is a complex pure metacyclic field of degree $20$ with signature $(0,10)$ whose Galois group

[TABLE]

is the semidirect product of two cyclic groups, isomorphic to SmallGroup$(20,3)$ [3, 4]. The action of the automorphisms is given by $\sigma(\sqrt[5]{D})=\zeta\cdot\sqrt[5]{D}$, $\sigma(\zeta)=\zeta$, $\tau(\sqrt[5]{D})=\sqrt[5]{D}$, $\tau(\zeta)=\zeta^{2}$.

The cyclotomic field $K=\mathrm{Fix}(\langle\sigma\rangle)$ is a complex cyclic quartic field and contains the real quadratic field $\mathbb{Q}(\sqrt{5})=\mathrm{Fix}(\langle\sigma,\tau^{2}\rangle)$ as its maximal real subfield $K^{+}=\mathbb{Q}(\zeta+\zeta^{-1})$. Consequently, there exists a real intermediate field $M=\mathbb{Q}(\sqrt[5]{D},\sqrt{5})$ of degree $10$ between $L$ and $N$, which is non-Galois with signature $(2,4)$. The two quintets of non-Galois subfields of $N$ are

[TABLE]

2.1. Normalization of radicands

Let $p\in\mathbb{P}\setminus\{2\}$ be an odd prime number. With an integer $s\geq 1$, exponents $1\leq e_{j}\leq p-1$ for $1\leq j\leq s$, and pairwise distinct prime numbers $q_{1},\ldots,q_{s}\in\mathbb{P}$ (where $2$ and $p$ may be among them) let $D=\prod_{j=1}^{s}\,q_{j}^{e_{j}}$ be a $p$-th power free radicand.

Since we desire a bijective correspondence between $p$-th power free radicands and pairwise non-isomorphic pure number fields $L=\mathbb{Q}(\sqrt[p]{D})$ of degree $p$, we introduce the following concepts.

Definition 2.1**.**

For a $p$th power free radicand $D$, let $D_{k}:=\prod\{q\in\mathbb{P}\mid v_{q}(D)=k\}$ for $1\leq k\leq p-1$ be the homogeneous component of degree $k$ of $D$, where $v_{q}$ denotes the $q$-adic valuation $\mathbb{Q}^{\times}\to\mathbb{Z}$, $\prod_{\ell\in\mathbb{P}}\,\ell^{n_{\ell}}\mapsto n_{q}$. Then we have $D=\prod_{k=1}^{p-1}\,D_{k}^{k}$, where each $D_{k}$ is squarefree.

In contrast, we let $D^{(1)}:=D$, and for $2\leq k\leq p-1$ we construct $D^{(k)}$ by forming the $k$-th power $D^{k}$ of $D$ and reducing all occurring exponents modulo $p$. Then the minimal positive integer $D^{(k_{0})}:=\min\{D^{(k)}\mid 1\leq k\leq p-1\}$ is called the normalized radicand of the field $L=\mathbb{Q}(\root p \of{D})$ and all the bigger integers $D^{(k)}$ with $k\neq k_{0}$ are called the co-radicands.

Example 2.1**.**

Only in the cubic case $p=3$, where $D^{(1)}=D_{1}D_{2}^{2}$ and $D^{(2)}=D_{1}^{2}D_{2}^{4}/D_{2}^{3}=D_{1}^{2}D_{2}$, we can easily achieve the normalization by selecting two coprime integers with $\gcd(D_{1},D_{2})=1$ such that the quadratic component $D_{2}<D_{1}$ is smaller than the linear component. In this case we can definitely say that $D^{(1)}$ is normalized and $D^{(2)}$ is the co-radicand, since $D_{1}D_{2}^{2}<D_{1}^{2}D_{2}$.

In the quintic case $p=5$, where the principal focus will lie in this work, we have the following four radicands, one of which is normalized:

[TABLE]

Note that generally

[TABLE]

Proposition 2.1**.**

There is a one-to-one correspondence between isomorphism classes of pure fields $L=\mathbb{Q}(\root p \of{D})$ of degree $p$ and normalized $p$-th power free radicands $D$.

Proof.

Any pure number field $L=\mathbb{Q}(\root p \of{R})$ of degree $p$ over $\mathbb{Q}$ can be generated by adjoining to the rational number field $\mathbb{Q}$ the unique real solution $\root p \of{R}$ of a pure equation $X^{p}-R=0$ with $R\in\mathbb{Q}^{\times}$ and $R\not\in(\mathbb{Q}^{\times})^{p}$. By multiplication of the rational radicand $R$ by the $p$th power of an integer, we can achieve that $R=D$ is a $p$th power free integer radicand as defined at the beginning of this section. Without loss of generality, we may assume that $D=D^{(1)}$ is the normalized radicand of the field $L=\mathbb{Q}(\root p \of{D})$. Then the powers $(\root p \of{D})^{k}=\root p \of{D^{k}}$ with exponents $2\leq k\leq p-1$ of the radical $\root p \of{D}$ are also elements of $L$. According to Formula (2.4), each of the co-radicals $\root p \of{D^{(k)}}=\root p \of{\frac{D^{k}}{n^{p}}}=\frac{1}{n}\cdot\root p \of{D^{k}}$ with some $n\in\mathbb{Z}\setminus\{0\}$ is also an element of $L$. Different normalized radicands are never multiplicatively congruent modulo $\mathbb{Z}^{p}$, but we have isomorphisms between all fields generated by co-radicals

[TABLE]

2.2. Conductor and discriminants

Let $R=q_{1}\cdots q_{s}$ be the squarefree product of all prime divisors of the radicand $D$ of the pure quintic field $L=\mathbb{Q}(\root 5 \of{D})$. Independently of the exponents $e_{1},\ldots,e_{s}$, the conductor $f$ of the cyclic quintic relative extension $N/K$ [20, Thm. 1, p. 103] is given by

[TABLE]

It is well known that the cyclotomic discriminant takes the value $d_{K}=+5^{3}=125$, and Hilbert’s Theorem $39$ on discriminants of composite fields shows the following result ([20, pp. 103–104]).

Theorem 2.1**.**

The metacyclic discriminant is given by

[TABLE]

the intermediate discriminant is given by

[TABLE]

and the pure quintic discriminant is given by

[TABLE]

Example 2.2**.**

The smallest radicands $D$ of pure quintic fields $L=\mathbb{Q}(\root 5 \of{D})$ with a given species and increasing number of prime divisors are

• •

$2$, $6=2\cdot 3$, $42=2\cdot 3\cdot 7$, for fields of the first species with $\gcd(D,5)=1$ (species 1b),

• •

$5$, $10=2\cdot 5$, $30=2\cdot 3\cdot 5$, for fields of the first species with $5\mid D$ (species 1a),

• •

$7$, $18=2\cdot 3^{2}$, $126=2\cdot 7\cdot 3^{2}$, for fields of the second species.

3. Differential factors and norm kernels

3.1. Ambiguous ideals

Let $E/F$ be a relative extension of algebraic number fields with relative degree $d:=[E:F]$. We do not assume that $E/F$ is a normal extension, and thus the law of decomposition of a prime ideal $\mathfrak{p}\in\mathbb{P}_{F}$ of $F$ in $E$ may be arbitrary:

[TABLE]

Since a Galois group $\mathrm{Gal}(E/F)$ of automorphisms may be missing, we cannot speak about ideals in $E$ which are invariant with respect to $F$. However, as a compensation, we define ambiguous ideals of $E$ relative to $F$.

Definition 3.1**.**

An ideal $\mathfrak{A}\in\mathcal{I}_{E}$ is called ambiguous with respect to $F$ (symbol $\mathfrak{A}\in\mathcal{I}_{E/F}$) if there exists a positive integer $n\in\mathbb{N}$ such that $\mathfrak{A}^{n}\in\mathcal{I}_{F}$, that is, $\mathfrak{A}\cdot\mathcal{I}_{F}$ has finite order in $\mathcal{I}_{E}/\mathcal{I}_{F}$.

Lemma 3.1**.**

Let $\mathfrak{A}\in\mathcal{I}_{E}$ be an ideal of $E$, then the following assertions are equivalent [18]:

1. (1)

$(\exists\,n\in\mathbb{N})\,\mathfrak{A}^{n}\in\mathcal{I}_{F}$, that is, $\mathfrak{A}\in\mathcal{I}_{E/F}$, 2. (2)

$(\exists\,e\mid d)\,\mathfrak{A}^{e}\in\mathcal{I}_{F}$, 3. (3)

$\mathfrak{A}^{d}\in\mathcal{I}_{F}$, 4. (4)

$\mathfrak{A}^{d}=N_{E/F}(\mathfrak{A})$.

Problem 3.1**.**

The following two questions can be answered by a semi-local investigation with respect to the prime ideals $\mathfrak{p}\in\mathbb{P}_{F}$ of the base field $F$ (whereas a local consideration concerns the prime ideals $\mathfrak{P}\in\mathbb{P}_{E}$ of the extension $E$):

1. (1)

Which fractional ideals $\mathfrak{A}\in\mathcal{I}_{E}$ are of finite order relative to $\mathcal{I}_{F}$? 2. (2)

Which fractional ideals $\mathfrak{A}\in\mathcal{I}_{E}$ become trivial under the relative norm $N_{E/F}$?

The first, respectively second, question of Problem 3.1 will be answered by Theorem 3.1, respectively Theorem 3.4.

Theorem 3.1**.**

Let $\mathfrak{A}\in\mathcal{I}_{E}$ be a fractional ideal of $E$. Then

[TABLE]

with exponents $v(\mathfrak{p})\in\mathbb{Z}$.

Proof.

(of Lemma 3.1 and Theorem 3.1) If $(\exists\,e\mid d)\,\mathfrak{A}^{e}\in\mathcal{I}_{F}$, that is, $d=e\cdot c$ for some $c\in\mathbb{N}$, then $\mathfrak{A}^{d}=(\mathfrak{A}^{e})^{c}\in\mathcal{I}_{F}$. Conversely, we trivially have $d\mid d$. The proof that the condition $(\exists\,n\in\mathbb{N})\,\mathfrak{A}^{n}\in\mathcal{I}_{F}$ implies $n\mid d$ is contained in the following proof of Theorem 3.1.

We select an arbitrary prime ideal $\mathfrak{p}\in\mathbb{P}_{F}$ of the base field $F$ and consider the condition $\mathfrak{A}^{n}\in\mathcal{I}_{F}$ semi-locally with respect to $\mathfrak{p}$: The $\mathfrak{p}$-component of $\mathfrak{A}^{n}$ lies in $\mathcal{I}_{F}$ if and only if there exists some exponent $m\in\mathbb{Z}$ such that $(\prod_{i=1}^{g}\,\mathfrak{P}_{i}^{v_{i}})^{n}=\mathfrak{p}^{m}$, where $v_{i}:=v_{\mathfrak{P}_{i}}(\mathfrak{A})$ for each $1\leq i\leq g$. This is equivalent with $(\prod_{i=1}^{g}\,\mathfrak{P}_{i}^{v_{i}})^{n}=(\prod_{i=1}^{g}\,\mathfrak{P}_{i}^{e_{i}})^{m}$, respectively, $\prod_{i=1}^{g}\,\mathfrak{P}_{i}^{v_{i}n}=\prod_{i=1}^{g}\,\mathfrak{P}_{i}^{e_{i}m}$. We obtain an equivalent system of linear equations with integer coefficients in the unknown integers $v_{i}$,

[TABLE]

which we can divide by the greatest common divisor $e$ of $e_{1},\ldots,e_{g}$,

[TABLE]

A solution of this system is given by $v_{i}:=\frac{e_{i}}{e}$ and $m:=\frac{n}{e}$. The minimal possible semi-local choice is $n:=e$, whence $m=1$. Finally, we can combine the semi-local solutions to a global solution by putting $n$ equal to the least common multiple $\mathrm{lcm}\{e(\mathfrak{p})\mid\mathfrak{p}\in\mathbb{P}_{F}\}$ of all semi-local values of $e=e(\mathfrak{p})$, which is still a divisor of $d$, since $d$ is a common multiple of these values. ∎

3.2. Primitive ambiguous ideals

There exists an infinitude of ambiguous ideals in $E$ relative to $F$, since for instance the infinitely many inert prime ideals are all ambiguous. We are interested in ambiguous ideals which are not contained in the base field and therefore we define primitivity.

Definition 3.2**.**

An element $\mathfrak{A}\cdot\mathcal{I}_{F}$ of the quotient group $\mathcal{I}_{E}/\mathcal{I}_{F}$ is called a primitive ideal of $E$ relative to $F$.

Corollary 3.1**.**

A primitive ambiguous ideal of $E$ relative to $F$ possesses the shape

[TABLE]

with exponents $v(\mathfrak{p})\in\mathbb{Z}/e(\mathfrak{p})\mathbb{Z}$.

Proof.

For $e(\mathfrak{p})=1$, we have $\prod_{i=1}^{g(\mathfrak{p})}\,\mathfrak{P}^{\frac{e_{i}}{e(\mathfrak{p})}}=\prod_{i=1}^{g(\mathfrak{p})}\,\mathfrak{P}^{e_{i}}=\mathfrak{p}\mathcal{O}_{E}\in\mathcal{I}_{F}$. This reduces the components of a primitive ambiguous ideal to the finitely many prime ideals $\mathfrak{p}\in\mathbb{P}_{F}$ with $e(\mathfrak{p})>1$. However, for $e(\mathfrak{p})>1$ and $v(\mathfrak{p})=e(\mathfrak{p})$, we have $(\prod_{i=1}^{g(\mathfrak{p})}\,\mathfrak{P}^{\frac{e_{i}}{e(\mathfrak{p})}})^{v(\mathfrak{p})}=\prod_{i=1}^{g(\mathfrak{p})}\,\mathfrak{P}^{e_{i}}=\mathfrak{p}\mathcal{O}_{E}\in\mathcal{I}_{F}$. Thus we can restrict the exponents of the finitely many components to finitely many values $v(\mathfrak{p})\in\mathbb{Z}/e(\mathfrak{p})\mathbb{Z}$. ∎

If $N/F$ with $F\leq E\leq N$ is the normal closure of $E/F$ with Galois group $G:=\mathrm{Gal}(N/F)$, then we set $\mathcal{I}_{E/F}:=\mathcal{I}_{E}\cap\mathcal{I}_{N/F}$, where $\mathcal{I}_{N/F}:=\{\mathfrak{A}\in\mathcal{I}_{N}\mid(\forall\,\phi\in G)\,\mathfrak{A}^{\phi}=\mathfrak{A}\}$.

Corollary 3.2**.**

$\mathcal{I}_{E/F}/\mathcal{I}_{F}\simeq\prod_{\mathfrak{p}\in\mathbb{P}_{F}}\,\mathbb{Z}/e(\mathfrak{p})\mathbb{Z}$*  is finite with  $\#(\mathcal{I}_{E/F}/\mathcal{I}_{F})=\prod_{\mathfrak{p}\in\mathbb{P}_{F}}\,e(\mathfrak{p})$.*

Proof.

This is an immediate consequence of Corollary 3.1. ∎

Remark 3.1**.**

If the relative degree $d=[E:F]$ is a prime number $q\in\mathbb{P}$, then the condition $e=e(\mathfrak{p})>1$ is only satisfied by prime ideals $\mathfrak{p}\in\mathbb{P}_{F}$ which ramify totally in $E$, since $e\mid d$ with $e>1$ and $d=q$ is only possible for $e=q$ and thus necessarily $f=g=1$. Therefore, if $T:=\#\{\mathfrak{p}\in\mathbb{P}_{F}\mid e(\mathfrak{p})>1\}$, then we have

[TABLE]

3.3. Relative different

As the title of the present paper suggests, our main goal is the investigation of differential principal factors, that is, principal ideal divisors $(A)=A\mathcal{O}_{N}\in\mathcal{P}_{N}$ of the relative different $\mathfrak{D}_{N/K}$ of pure metacyclic fields $N=\mathbb{Q}(\root p \of{D},\zeta)$ with respect to the cyclotomic subfield $K=\mathbb{Q}(\zeta)$. For this purpose, a brief excursion to the general theory will be illuminating.

Let $E/F$ be an extension of algebraic number fields. We fix a prime ideal $\mathfrak{P}\in\mathbb{P}_{E}$ of $E$ and we denote the ramification exponent of $\mathfrak{P}$ over the prime ideal $\mathfrak{p}:=F\cap\mathfrak{P}\in\mathbb{P}_{F}$ of $F$ below $\mathfrak{P}$ by $e:=v_{\mathfrak{P}}(\mathfrak{p})\in\mathbb{N}$.

Theorem 3.2**.**

The contribution of the prime ideal $\mathfrak{P}\in\mathbb{P}_{E}$ to the relative different $\mathfrak{D}_{E/F}\in\mathcal{I}_{E}$ of $E/F$ depends on the ramification of $\mathfrak{P}$ in $E/F$ and is given by

[TABLE]

The relative discriminant $\mathfrak{d}_{E/F}\in\mathcal{I}_{F}$ of $E/F$ is the relative norm of the relative different,

[TABLE]

Proof.

Formula 3.5 is proved in [24, Thm. 2.6, pp. 199-200] and formula 3.6 is proved in [24, Thm. 2.9, pp. 201-202]. Note that, in particular, $v_{\mathfrak{P}}(\mathfrak{D}_{E/F})=0$ $\Longleftrightarrow$ $e=1$ ($\mathfrak{P}$ unramified). ∎

Now we apply the general theory to our special situations, either the Kummer extension $E:=N=\mathbb{Q}(\root p \of{D},\zeta)$ of the cyclotomic base field $F:=K=\mathbb{Q}(\zeta)$ or the pure extension $E:=L=\mathbb{Q}(\root p \of{D})$ of the rational base field $F:=\mathbb{Q}$. We begin with results for the quintic case $p=5$.

Theorem 3.3**.**

Let $q\in\mathbb{P}$ be a prime number and denote by $\mathfrak{Q}\in\mathbb{P}_{N}$ a prime ideal of $N=\mathbb{Q}(\root 5 \of{D},\zeta)$ dividing the extension ideal $q\mathcal{O}_{N}$. As before, $f$ denotes the conductor of $N/K$. Then the contribution of $\mathfrak{Q}$ to the relative different $\mathfrak{D}_{N/K}$ is given by

[TABLE]

Now denote by $\mathfrak{Q}\in\mathbb{P}_{L}$ a prime ideal of $L=\mathbb{Q}(\root 5 \of{D})$ dividing the extension ideal $q\mathcal{O}_{L}$. Then the contribution of $\mathfrak{Q}$ to the absolute different $\mathfrak{D}_{L/\mathbb{Q}}$ is given by

[TABLE]

Remark 3.1 together with Theorem 3.3 explains the designation differential factors for primitive ambiguous ideals $\mathfrak{A}\in\mathcal{I}_{N}$ and $\mathfrak{a}\in\mathcal{I}_{L}$ which are not divisible by a prime ideal above $5$:

$\mathfrak{A}\cdot\mathcal{I}_{K}\in\mathcal{I}_{N/K}/\mathcal{I}_{K}$ $\Longleftrightarrow$ $\mathfrak{A}\mid\mathfrak{R}_{N/K}^{4}$ $\Longleftrightarrow$ $\mathfrak{A}\mid\mathfrak{D}_{N/K}$, $\mathfrak{a}\cdot\mathcal{I}_{\mathbb{Q}}\in\mathcal{I}_{L/\mathbb{Q}}/\mathcal{I}_{\mathbb{Q}}$ $\Longleftrightarrow$ $\mathfrak{a}\mid\mathfrak{R}_{L/\mathbb{Q}}^{4}$ $\Longleftrightarrow$ $\mathfrak{a}\mid\mathfrak{D}_{L/\mathbb{Q}}$.

3.4. Kernel of the relative norm

In Theorem 3.4 we shall find the answer to the second question of Problem 3.1.

Theorem 3.4**.**

Let $\mathfrak{A}\in\mathcal{I}_{E}$ be a fractional ideal of $E$. Then

[TABLE]

Proof.

Let $\mathfrak{p}\in\mathbb{P}_{F}$ be a prime ideal of the base field $F$. We consider the condition $N_{E/F}(\mathfrak{A})=\mathcal{O}_{F}$ semi-locally with respect to $\mathfrak{p}$: $0=v_{\mathfrak{p}}(\mathcal{O}_{F})=v_{\mathfrak{p}}(N_{E/F}(\mathfrak{A}))=v_{\mathfrak{p}}(N_{E/F}(\prod_{i=1}^{g}\,\mathfrak{P}_{i}^{v_{i}}))=v_{\mathfrak{p}}(\prod_{i=1}^{g}\,N_{E/F}(\mathfrak{P}_{i})^{v_{i}})=v_{\mathfrak{p}}(\prod_{i=1}^{g}\,(\mathfrak{p}^{f_{i}})^{v_{i}})=v_{\mathfrak{p}}(\mathfrak{p}^{\sum_{i=1}^{g}\,f_{i}v_{i}})=\sum_{i=1}^{g}\,f_{i}v_{i}$, where $v_{i}=v_{\mathfrak{P}_{i}}(\mathfrak{A})$. ∎

Note that, in the preceding proof, $\prod_{i=1}^{g}\,\mathfrak{P}_{i}^{v_{i}}$ is only the semi-local component of $\mathfrak{A}$ with respect to $\mathfrak{p}$. Other components of $\mathfrak{A}$ do not contribute to the valuation $v_{\mathfrak{p}}$.

Corollary 3.3**.**

1. (1)

If $N_{E/F}(\mathfrak{A})=\mathcal{O}_{F}$, then only prime ideals $\mathfrak{p}\in\mathbb{P}_{F}$ which split in $E$ can contribute to $\mathfrak{A}=\prod_{g(\mathfrak{p})\geq 2}\,\prod_{\mathfrak{P}\mid\mathfrak{p}}\,\mathfrak{P}^{v_{\mathfrak{P}}(\mathfrak{A})}$. 2. (2)

If $\mathfrak{A}$ is an integral ideal of $E$, then $N_{E/F}(\mathfrak{A})=\mathcal{O}_{F}\ \Longleftrightarrow\ \mathfrak{A}=\mathcal{O}_{E}$. 3. (3)

If $E/F$ is a quadratic extension, then $N_{E/F}(\mathfrak{A})=\mathcal{O}_{F}\ \Longleftrightarrow\ \mathfrak{A}=\prod_{g(\mathfrak{p})=2}\,(\mathfrak{P}_{1}\mathfrak{P}_{2}^{-1})^{v_{\mathfrak{P_{1}}}(\mathfrak{A})}$.

Proof.

1. (1)

For non-split prime ideals $\mathfrak{p}\in\mathbb{P}_{F}$ with $g(\mathfrak{p})=1$, the sum $\sum_{\mathfrak{P}\mid\mathfrak{p}}\,f(\mathfrak{P}/\mathfrak{p})\cdot v_{\mathfrak{P}}(\mathfrak{A})$ degenerates to a single summand $f(\mathfrak{P}/\mathfrak{p})\cdot v_{\mathfrak{P}}(\mathfrak{A})=0$, which implies $v_{\mathfrak{P}}(\mathfrak{A})=0$. 2. (2)

If $\mathfrak{A}$ is integral, then all $v_{\mathfrak{P}}(\mathfrak{A})\geq 0$ are non-negative. Consequently, the sum $\sum_{\mathfrak{P}\mid\mathfrak{p}}\,f(\mathfrak{P}/\mathfrak{p})\cdot v_{\mathfrak{P}}(\mathfrak{A})$ with positive $f(\mathfrak{P}/\mathfrak{p})\geq 1$ can be zero only for constant $v_{\mathfrak{P}}(\mathfrak{A})=0$. 3. (3)

If $d=[E:F]=2$, then the necessary condition $g(\mathfrak{p})\geq 2$ of item (1) implies $g(\mathfrak{p})=2$, $\mathfrak{p}\mathcal{O}_{E}=\mathfrak{P}_{1}\mathfrak{P}_{2}$, $f(\mathfrak{P_{i}}/\mathfrak{p})=1$, and $\sum_{i=1}^{2}\,v_{\mathfrak{P_{i}}}(\mathfrak{A})=0$, that is, $v_{\mathfrak{P_{2}}}(\mathfrak{A})=-v_{\mathfrak{P_{1}}}(\mathfrak{A})$.∎

4. Dimensions of spaces of differential factors

4.1. Absolute differential factors

Let $T$ be the number of primes $q_{1},\ldots,q_{T}$ dividing the conductor of the cyclic quintic extension $N/K$, and $\mathfrak{q}_{1},\ldots,\mathfrak{q}_{T}$ their overlying prime ideals in $L$.

Theorem 4.1**.**

The $\mathbb{F}_{5}$-vectorspace $\mathcal{I}_{L/\mathbb{Q}}/\mathcal{I}_{\mathbb{Q}}$ of absolute ambiguous ideals has the dimension

[TABLE]

which is finite but unbounded. A basis representation is given by

[TABLE]

Proof.

This follows from formulas (3.3) and (3.4) by taking $d=q=5$. ∎

Corollary 4.1**.**

The $\mathbb{F}_{5}$-subspace $\mathcal{P}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}\leq\mathcal{I}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}$ of absolute ambiguous principal ideals has the dimension

[TABLE]

which is finite and bounded by

[TABLE]

In particular, the DPF type $\gamma$ with maximal value $A=3$ can occur for $T\geq 3$ only.

Proof.

The bound in formula (4.4) is obtained by combining Theorem 6.1 with formula (4.1). ∎

4.2. Intermediate and relative differential factors

We start with a general statement concerning groups of primitive ambiguous ideals.

Theorem 4.2**.**

Let $p$ be a prime and $q>1$ be an integer coprime to $p$. Suppose $E_{0}/F_{0}$ is a number field extension of degree $q$, $F/F_{0}$ is an extension of degree $p$, and $E=F\cdot E_{0}$ is the compositum of $F$ and $E_{0}$.

Then the norm homomorphism $N_{E/F}:\,\mathcal{I}_{E}\to\mathcal{I}_{F}$ satisfies

[TABLE]

and induces an epimorphism

[TABLE]

There are isomorphisms of elementary abelian $p$-groups

[TABLE]

and

[TABLE]

Proof.

The image of an ambiguous ideal $\mathfrak{A}\in\mathcal{I}_{E/E_{0}}$ under the norm $N_{E/F}$ is an ambiguous ideal in $\mathcal{I}_{F/F_{0}}$: according to item (3) and (4) of Lemma 3.1, we have $\mathfrak{A}^{p}=\mathfrak{n}:=N_{E/E_{0}}(\mathfrak{A})$, since $[E:E_{0}]=[(F\cdot E_{0}):E_{0}]=[F:(F\cap E_{0})]=[F:F_{0}]=p$, and therefore $N_{E/F}(\mathfrak{A})^{p}=N_{E/F}(\mathfrak{A}^{p})=N_{E/F}(\mathfrak{n})=N_{E/F}(N_{E/E_{0}}(\mathfrak{A}))=N_{E_{0}/F_{0}}(N_{E/E_{0}}(\mathfrak{A}))=N_{E/F_{0}}(\mathfrak{A})=N_{F/F_{0}}(N_{E/F}(\mathfrak{A}))\in\mathcal{I}_{F_{0}}$.

So, $N_{E/F}:\,\mathcal{I}_{E}\to\mathcal{I}_{F}$ restricts to $N_{E/F}:\,\mathcal{I}_{E/E_{0}}\to\mathcal{I}_{F/F_{0}}$, but the restriction is not surjective: if $\mathfrak{L}\in\mathcal{I}_{E/E_{0}}$ is an inert prime ideal of $E/F$, then $N_{E/F}(\mathfrak{L})=\mathfrak{l}^{q}$ and thus $\mathfrak{l}=\mathfrak{L}\cap\mathcal{O}_{F}\not\in\mathrm{im}(N_{E/F})$.

However, generally we have $N_{E/F}(\mathcal{I}_{E_{0}})=N_{E_{0}/F_{0}}(\mathcal{I}_{E_{0}})\leq\mathcal{I}_{F_{0}}$ and there exists an induced mapping $N_{E/F}:\,\mathcal{I}_{E/E_{0}}/\mathcal{I}_{E_{0}}\to\mathcal{I}_{F/F_{0}}/\mathcal{I}_{F_{0}}$ which is an epimorphism, since $\gcd(p,q)=1$, respectively $(\exists\,a,b\in\mathbb{Z})\,ap+bq=1$, and thus $N_{E/F}(\mathfrak{L}^{b})=\mathfrak{l}^{bq}=\mathfrak{l}\cdot\mathfrak{l}^{-ap}\equiv\mathfrak{l}\,(\mathrm{mod}\,\mathcal{I}_{F_{0}})$, since $\mathfrak{l}^{p}\in\mathcal{I}_{F_{0}}$.

The isomorphism theorem yields a quotient representation $\mathcal{I}_{F/F_{0}}/\mathcal{I}_{F_{0}}\simeq(\mathcal{I}_{E/E_{0}}/\mathcal{I}_{E_{0}})/\ker(N_{E/F})$, and since the groups of primitive ambiguous ideals $\mathcal{I}_{F/F_{0}}/\mathcal{I}_{F_{0}}$ and $\mathcal{I}_{E/E_{0}}/\mathcal{I}_{E_{0}}$ are elementary abelian $p$-groups (for instance, $\mathfrak{A}\in\mathcal{I}_{E/E_{0}}$ implies $\mathfrak{A}^{p}=N_{E/E_{0}}(\mathfrak{A})\in\mathcal{I}_{E_{0}}$), there is an equivalent direct product representation $\mathcal{I}_{E/E_{0}}/\mathcal{I}_{E_{0}}\simeq(\mathcal{I}_{F/F_{0}}/\mathcal{I}_{F_{0}})\times\ker(N_{E/F})$. ∎

Let $s_{2}$, resp. $s_{4}$, be the number of prime divisors $\ell_{1},\ldots,\ell_{s_{2}}\equiv-1\,(\mathrm{mod}\,5)$, resp. $\ell_{s_{2}+1},\ldots,\ell_{s_{2}+s_{4}}\equiv+1\,(\mathrm{mod}\,5)$, of the conductor $f$ of $N/K$, which split in $M$. Denote by $\mathcal{L}_{1},\mathcal{L}_{1}^{\tau},\ldots,\mathcal{L}_{s_{2}+s_{4}},\mathcal{L}_{s_{2}+s_{4}}^{\tau}$ their overlying prime ideals in $M$.

Theorem 4.3**.**

The $\mathbb{F}_{5}$-vectorspace $(\mathcal{I}_{M/K^{+}}/\mathcal{I}_{K^{+}})\bigcap\ker(N_{M/L})$ of intermediate ambiguous ideals has the dimension

[TABLE]

which is finite but unbounded. A basis representation is given by

[TABLE]

where $\mathcal{K}^{(\ell_{i})}=\mathcal{L}_{i}^{1+4\tau}=\mathcal{L}_{i}\cdot(\mathcal{L}_{i}^{\tau})^{4}$, for each $1\leq i\leq s_{2}+s_{4}$.

Proof.

The formulas (3.3) and (3.4) with substitutions $d=q\mapsto 5$ and $T\mapsto T+s_{2}+s_{4}$ imply $\dim_{\mathbb{F}_{5}}(\mathcal{I}_{M/K^{+}}/\mathcal{I}_{K^{+}})=T+s_{2}+s_{4}$. Consequently, Theorem 4.2 yields formula (4.5) and Theorem 3.4 together with item (3) of Corollary 3.3 yields formula (4.6). ∎

Corollary 4.2**.**

The $\mathbb{F}_{5}$-subspace $(\mathcal{P}_{M/K^{+}}/\mathcal{P}_{K^{+}})\bigcap\ker(N_{M/L})\leq(\mathcal{I}_{M/K^{+}}/\mathcal{P}_{K^{+}})\bigcap\ker(N_{M/L})$ of intermediate ambiguous principal ideals has the dimension

[TABLE]

which is finite and bounded by

[TABLE]

In particular, the DPF type $\alpha_{3}$ with maximal value $I=2$ can occur for $s_{2}+s_{4}\geq 2$ only.

Proof.

The bound in formula (4.8) is obtained by combining Theorem 6.1 with formula (4.5). ∎

Finally, denote by $\mathfrak{L}_{s_{2}+1},\mathfrak{L}_{s_{2}+1}^{\tau^{2}},\mathfrak{L}_{s_{2}+1}^{\tau},\mathfrak{L}_{s_{2}+1}^{\tau^{3}},\ldots,\mathfrak{L}_{s_{2}+s_{4}},\mathfrak{L}_{s_{2}+s_{4}}^{\tau^{2}},\mathfrak{L}_{s_{2}+s_{4}}^{\tau},\mathfrak{L}_{s_{2}+s_{4}}^{\tau^{3}}$ the overlying prime ideals in $N$ of the prime divisors $\ell_{s_{2}+1},\ldots,\ell_{s_{2}+s_{4}}\equiv+1\,(\mathrm{mod}\,5)$ of the conductor $f$ of $N/K$ which $2$-split in $M$ and $4$-split in $N$.

Theorem 4.4**.**

The $\mathbb{F}_{5}$-vectorspace $(\mathcal{I}_{N/K}/\mathcal{I}_{K})\bigcap\ker(N_{N/M})$ of relative ambiguous ideals has the dimension

[TABLE]

which is finite but unbounded. A basis representation with generators of $\tau$-invariant $1$-dimensional $\mathbb{F}_{5}$-subspaces is given by

[TABLE]

where $\mathfrak{K}_{1}^{(\ell_{i})}=\mathfrak{L}_{i}^{1+4\tau^{2}+2\tau+3\tau^{3}}$ and $\mathfrak{K}_{2}^{(\ell_{i})}=\mathfrak{L}_{i}^{1+4\tau^{2}+3\tau+2\tau^{3}}$, for each $s_{2}+1\leq i\leq s_{2}+s_{4}$.

Proof.

The formulas (3.3) and (3.4) with substitutions $d=q\mapsto 5$ and $T\mapsto T+s_{2}+3s_{4}$ imply $\dim_{\mathbb{F}_{5}}(\mathcal{I}_{N/K}/\mathcal{I}_{K})=T+s_{2}+3s_{4}$. Consequently, Theorem 4.2 yields formula (4.9). But Theorem 3.4 does not yield formula (4.10) with generators of $1$-dimensional $\tau$-invariant subspaces. Here, we must use Proposition 7.1. ∎

Corollary 4.3**.**

The $\mathbb{F}_{5}$-subspace $(\mathcal{P}_{N/K}/\mathcal{P}_{K})\bigcap\ker(N_{N/M})\leq(\mathcal{I}_{N/K}/\mathcal{P}_{K})\bigcap\ker(N_{N/M})$ of relative ambiguous principal ideals has the dimension

[TABLE]

which is finite and bounded by

[TABLE]

In particular, the DPF type $\alpha_{1}$ with maximal value $R=2$ can occur for $s_{4}\geq 1$ only.

Proof.

The bound in formula (4.12) is obtained by combining Theorem 6.1 with formula (4.9). ∎

5. Class number relations and index of subfield units

In $1973$, Charles J. Parry has determined the class number relation between $h_{N}=\#\mathrm{Cl}(N)$ and $h_{L}=\#\mathrm{Cl}(L)$ for a pure metacyclic field $N=\mathbb{Q}(\zeta_{5},\sqrt[5]{D})$ with pure quintic subfield $L=\mathbb{Q}(\sqrt[5]{D})$:

[TABLE]

[27, Thm. I, p. 476], where $U_{N}$ denotes the unit group of $N$, $U_{0}$ is the subgroup of $U_{N}$ generated by all units of the conjugate fields $L_{j}=\mathbb{Q}(\zeta_{5}^{j}\cdot\root 5 \of{D})$, $0\leq j\leq 4$, of $L$ and of $K$, and the index of subfield units $(U_{N}:U_{0})$ can take seven possible values $5^{E}$ with $0\leq E\leq 6$ [27, Thm. II, p. 478].

Parry’s class number relation is the special case $p=5$ of the following general class number formula by Colin D. Walter.

Theorem 5.1**.**

Let $p$ be an odd prime. For a $p$-th power free integer $D\geq 2$ let $N=\mathbb{Q}(\zeta_{p},\sqrt[p]{D})$ be a pure metacyclic field of degree $p(p-1)$ with pure subfield $L=\mathbb{Q}(\sqrt[p]{D})$ of degree $p$. Denote by $U_{N}$ the unit group of $N$, and by $U_{0}$ the subgroup of $U_{N}$ generated by all units of conjugate fields $L_{j}=\mathbb{Q}(\zeta_{p}^{j}\cdot\root p \of{D})$, $0\leq j\leq p-1$, of $L$ and of $K$. Then the following class number formula holds:

[TABLE]

Proof.

This general class number formula is a consequence of results on Frobenius extensions by Walter [34, Theorem 3.6, p. 222, and Theorem 4.4, p. 223], who applied the results to pure metacyclic extensions in [35, (1.7), (1.9), p. 4]. ∎

Example 5.1**.**

Anticipating some of our numerical results in section 10 and the types in section 6, we give the smallest radicands $D$ of pure quintic fields $L=\mathbb{Q}(\sqrt[5]{D})$ where the various values of the exponent $e$ actually occur:

• •

$E=6$ for $D=6=2\cdot 3$ of type $\gamma$,

• •

$E=5$ for $D=2$ of type $\varepsilon$,

• •

$E=4$ for $D=22=2\cdot 11$ of type $\beta_{2}$,

• •

$E=3$ for $D=11$ of type $\alpha_{2}$,

• •

$E=2$ for $D=31$ of type $\alpha_{1}$,

• •

$E=1$ for $D=33=3\cdot 11$ of type $\alpha_{2}$.

However, we point out that we did not find a realization of $E=0$, that is, obviously $U_{N}$ is never generated exclusively by subfield units.

6. Galois cohomology and Herbrand quotient of $U_{N}$

Let $p$ be an odd prime number, $D\geq 2$ a $p$-th power free integer, and $\zeta_{p}$ be a primitive $p$-th root of unity. Our classification of pure metacyclic fields $N=\mathbb{Q}(\zeta_{p},\sqrt[p]{D})$, which are cyclic Kummer extensions of the cyclotomic field $K=\mathbb{Q}(\zeta_{p})$ with relative automorphism group $G=\mathrm{Gal}(N/K)=\langle\sigma\rangle$, is based on the Galois cohomology of the unit group $U_{N}$ viewed as a $G$-module. The primary invariant is the group $H^{0}(G,U_{N})=U_{K}/N_{N/K}(U_{N})$ of order $p^{U}$ with $0\leq U\leq\frac{p-1}{2}$ which is related to the group $H^{1}(G,U_{N})=(U_{N}\cap\ker(N_{N/K}))/U_{N}^{1-\sigma}$ of order $p^{P}$ by the theorem on the Herbrand quotient of $U_{N}$,

[TABLE]

The secondary invariant is a natural decomposition of the group $H^{1}(G,U_{N})\simeq\mathcal{P}_{N/K}/\mathcal{P}_{K}$ of primitive ambiguous principal ideals of $N/K$, which can be viewed as principal ideals dividing the relative different $\mathfrak{D}_{N/K}$, and are therefore called differential principal factors (DPF) of $N/K$,

[TABLE]

where $L=\mathbb{Q}(\sqrt[p]{D})$ denotes the real non-Galois pure subfield of degree $p$ of $N$, the group of bottom DPF, $\mathcal{P}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}$, is of order $p^{B}$, and the group of top DPF, $(\mathcal{P}_{N/K}/\mathcal{P}_{K})\cap\ker(N_{N/L})$, of order $p^{T}$.

Based on the preceding preparation, we now state our Main Theorem on pure quintic fields $L=\mathbb{Q}(\sqrt[5]{D})$ and their Galois closure $N=\mathbb{Q}(\zeta_{5},\sqrt[5]{D})$, that is the case $p=5$. Since there exist intermediate extensions $\mathbb{Q}<K^{+}<K$ and $L<M<N$, namely the maximal real subfield $K^{+}=\mathbb{Q}(\sqrt{5})$ of $K$ and the real non-Galois subfield $M=\mathbb{Q}(\sqrt{5},\sqrt[5]{D})$ of absolute degree $10$ of $N$, the top DPF can be decomposed further (again using Theorem 4.2, as in formula (6.2))

[TABLE]

where the group of absolute DPF, $\mathcal{P}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}$, is of order $p^{A}$, the group of intermediate DPF, $(\mathcal{P}_{M/K^{+}}/\mathcal{P}_{K^{+}})\cap\ker(N_{M/L})$, is of order $p^{I}$, and the group of relative DPF, $(\mathcal{P}_{N/K}/\mathcal{P}_{K})\cap\ker(N_{N/M})$, is of order $p^{R}$. Note that $A=B$, $I+R=T$, and Theorem 6.1 coincides with Theorem 1.3.

Theorem 6.1**.**

Each pure metacyclic field $N=\mathbb{Q}(\zeta_{5},\sqrt[5]{D})$ of absolute degree $[N:\mathbb{Q}]=20$ with $5$-th power free radicand $D\in\mathbb{Z}$, $D\geq 2$, belongs to precisely one of the following $13$ differential principal factorization types, in dependence on the invariant $U$ and the triplet $(A,I,R)$.

The types $\delta_{1}$, $\delta_{2}$, $\varepsilon$ are characterized additionally by $\zeta_{5}\not\in N_{N/K}(U_{N})$, and the types $\zeta_{1}$, $\zeta_{2}$, $\eta$ by $\zeta_{5}\in N_{N/K}(U_{N})$.

Proof.

The claim is a consequence of combining formulas (6.1), (6.2), and (6.3). ∎

For the sake of comparison, we state the much more simple analogue for pure cubic fields $L=\mathbb{Q}(\sqrt[3]{D})$ and their Galois closure $N=\mathbb{Q}(\zeta_{3},\sqrt[3]{D})$, that is the case $p=3$.

Theorem 6.2**.**

Each pure metacyclic field $N=\mathbb{Q}(\zeta_{3},\sqrt[3]{D})$ of absolute degree $[N:\mathbb{Q}]=6$ with cube free radicand $D\in\mathbb{Z}$, $D\geq 2$, belongs to precisely one of the following $3$ differential principal factorization types, in dependence on the invariant $U$ and the pair $(B,T)$:

Proof.

A part of the proof is due to Barrucand and Cohn [2] who distinguished $4$ different types, $\mathrm{I}\hat{=}\beta$, $\mathrm{II}$, $\mathrm{III}\hat{=}\alpha$, and $\mathrm{IV}\hat{=}\gamma$. However, Halter-Koch [10] showed the impossibility of one of these types, namely type $\mathrm{II}$. Our new proof is the combination of formulas (6.1) and (6.2). ∎

6.1. Herbrand quotient of the units of $N/K$

In section 5, we have seen that the unit index $(U_{N}:U_{0})=5^{e}$ admits a coarse classification of pure quintic fields according to the seven possible values of the exponent $0\leq e\leq 6$. However, the Galois cohomology of the unit group $U_{N}$ with respect to the cyclic quintic Kummer extension $N/K$ provides additional structural information. The Herbrand quotient of $U_{N}$ under the action of cyclic group $G=\mathrm{Gal}(N/K)=\langle\sigma\rangle$ is defined by

[TABLE]

where the symbolic difference $\Delta:E\mapsto E^{\sigma-1}$ and the norm $\mathcal{N}:E\mapsto E^{1+\sigma+\cdots+\sigma^{4}}$ are endomorphisms of the unit group $U_{N}$ and the subgroup $E_{N/K}:=U_{N}\cap\ker(N_{N/K})$ consists of the units $E$ in $U_{N}$ with relative norm $N_{N/K}{E}=1$. According to Takagi ($1920$), Hasse ($1927$) and Herbrand ($1932$), the Herbrand quotient of the $G$-module $U_{N}$ has the value

[TABLE]

since Archimedean places do not yield a contribution, if $[N:K]$ is an odd prime.

6.2. Ambiguous principal ideals

Since the cyclotomic unit group $U_{K}$ is generated by $\langle-1,\eta,\zeta\rangle$, where $\eta>1$ denotes the fundamental unit of $K^{+}=\mathbb{Q}(\sqrt{5})$, and since $N_{N/K}(U_{N})\geq N_{N/K}(U_{K})=U_{K}^{5}=\langle-1,\eta^{5}\rangle$, we obtain the possible values of the unit norm index $(U_{K}:N_{N/K}(U_{N}))\in\{1,5,25\}$, according to whether $N_{N/K}(U_{N})$ contains $\{\eta,\zeta\}$ or only $\eta$, resp. $\zeta$, or none of them.

The somewhat abstract quotient $E_{N/K}/U_{N}^{\sigma-1}$ is isomorphic to the more ostensive quotient $\mathcal{P}_{r}:=\mathcal{P}_{N/K}/\mathcal{P}_{K}$ of the group of relative ambiguous principal ideals of $N/K$ modulo the subgroup of principal ideals of $K$, according to Iwasawa (or also to Hilbert’s Theorems $92$ and $94$), and by the Hasse theorem on the Herbrand quotient of the $G$-module $U_{N}$, we have

[TABLE]

Even in the worst case $(\mathcal{P}_{N/K}:\mathcal{P}_{K})=5$, there are at least the radicals $\root 5 \of{D},\ldots,\root 5 \of{D^{4}}$, and the unit $1$ which generate $5$ distinct absolute ambiguous principal ideals of $L/\mathbb{Q}$.

Example 6.1**.**

Anticipating some of our computational results in section § 10, we give the smallest radicands $D$ of pure quintic fields $L=\mathbb{Q}(\root 5 \of{D})$ where the various values of $\#\mathcal{P}_{r}=(\mathcal{P}_{N/K}:\mathcal{P}_{K})$ actually occur:

• •

$\#\mathcal{P}_{r}=5$ for $D=5$ of type $\vartheta$,

• •

$\#\mathcal{P}_{r}=25$ for $D=2$ of type $\varepsilon$,

• •

$\#\mathcal{P}_{r}=125$ for $D=6=2\cdot 3$ of type $\gamma$.

Generally, the fixed value of the Herbrand quotient for a given type of field extension $N/K$ can be interpreted by the following principle.

Trade-off Principle. If many units in $U_{K}$ can be represented as norms of units in $U_{N}$, then the extension $N/K$ contains only few ambiguous principal ideals.

If only few units in $U_{K}$ can be represented as norms of units in $U_{N}$, then the extension $N/K$ contains many ambiguous principal ideals.

6.3. Differential principal factorizations (DPF)

As opposed to the claim $\mathcal{P}_{N}\cap\mathcal{I}_{K}>\mathcal{P}_{K}$ of Hilbert’s Theorem $94$ for an unramified cyclic extension $N/K$ of prime degree with conductor $f=1$, the subgroup $1=\mathcal{P}_{N}\cap\mathcal{I}_{K}/\mathcal{P}_{K}<\mathcal{P}_{N/K}/\mathcal{P}_{K}$, the so-called capitulation kernel of $N/K$, is trivial for our ramified relative extension with $f>1$, because the cyclotomic field $K$ has class number $h_{K}=1$. However, the elementary abelian $5$-group $\mathcal{P}_{r}=\mathcal{P}_{N/K}/\mathcal{P}_{K}$, whose generators are principal ideals dividing the relative different of $N/K$, so-called differential principal factors, consists of three nested subgroups, $\mathcal{P}_{r}\geq\mathcal{P}_{i}\geq\mathcal{P}_{a}$ (similar to but not identical with our definitions in [19] and in § 4, since we only want to indicate another slightly different point of view),

• •

absolute DPF of $L/\mathbb{Q}$, $\mathcal{P}_{a}$, always containing the above mentioned radicals,

• •

intermediate DPF of $M|K^{+}$, $\mathcal{P}_{i}\setminus\mathcal{P}_{a}$, such that the proper cosets of $\mathcal{P}_{a}$ in $\mathcal{P}_{i}$ do not project down to $L/\mathbb{Q}$ by taking the norm $N_{M/L}$, and

• •

relative DPF of $N/K$, $\mathcal{P}_{r}\setminus\mathcal{P}_{i}$, such that the proper cosets of $\mathcal{P}_{i}$ in $\mathcal{P}_{r}$ do not even project down to $M/K^{+}$ by taking the norm $N_{N/M}$.

6.4. DPF types of pure quintic fields

We define thirteen possible differential principal factorization types of pure quintic number fields $L=\mathbb{Q}(\root 5 \of{D})$, according to the generators of the group $\mathrm{Norm}_{N|K}{U_{N}}$ as the primary invariant and the triplet $(A,I,R)$ defined by

the order $5^{A}:=\#\mathcal{P}_{a}$ of the group of absolute DPF of $L/\mathbb{Q}$,

the index $5^{I}:=(\mathcal{P}_{i}:\mathcal{P}_{a})$ in the group of intermediate DPF of $M/K^{+}$, and

the index $5^{R}:=(\mathcal{P}_{r}:\mathcal{P}_{i})$ in the group of relative DPF of $N/K$,

as the secondary invariant (similar to but not identical with our definitions in [19] and in § 4). The connection between the various quantities is given by the chain of equations

[TABLE]

Note that the index of subfield units $(U_{N}:U_{0})$ does not enter the definition of the DPF types.

7. Orthogonal idempotents

Firstly, let $G=\langle\tau\rangle\simeq C_{2}$ be the unique group of prime order $2$, which is cyclic. For subsequent applications, $G$ may be viewed as the relative group $\mathrm{Gal}(N/L)$ of the Galois closure $N=\mathbb{Q}(\root 3 \of{D},\zeta_{3})$ over a pure cubic field $L=\mathbb{Q}(\root 3 \of{D})$.

Lemma 7.1**.**

Let $\mu_{2}=\langle-1\rangle\simeq C_{2}$ be the group of square roots of unity, then the character group $G^{\ast}:=\mathrm{Hom}(G,\mu_{2})\simeq G$ of $G$ consists of two characters $\{1=\chi_{0},\chi_{1}\}$ such that $\chi_{j}(\tau)=-1^{j}$ for $0\leq j\leq 1$. The values of these characters for $g\in G$ are shown in Table 7.1.

Two central orthogonal idempotents in the group ring $R=\mathbb{Q}[G]$ are given by

[TABLE]

More explicitly, we have $\psi_{0}=\frac{1}{2}(1+\tau)$ and $\psi_{1}=\frac{1}{2}(1-\tau)$. They satisfy the relations $\psi_{0}+\psi_{1}=1$ and $\psi_{i}\cdot\psi_{j}=\delta_{i,j}\psi_{j}$ for $0\leq i,j\leq 1$.

For our application to the arithmetic of pure metacyclic fields $N=\mathbb{Q}(\root 3 \of{D},\zeta_{3})$ of degree $6$, we can view these central orthogonal idempotents as elements of the group ring $(\mathbb{Z}/3\mathbb{Z})[G]$ using the isomorphism $\mu_{2}=\langle-1\rangle\simeq C_{2}\simeq U(\mathbb{Z}/3\mathbb{Z})$ which arises by mapping $-1\mapsto\bar{2}$. Then we obtain:

[TABLE]

Secondly, let $G=\langle\tau\rangle\simeq C_{4}$ be the cyclic group of order $4$. For subsequent applications, $G$ may be viewed as the relative group $\mathrm{Gal}(N/L)$ of the Galois closure $N=\mathbb{Q}(\root 5 \of{D},\zeta_{5})$ over a pure quintic field $L=\mathbb{Q}(\root 5 \of{D})$.

Lemma 7.2**.**

Let $\mu_{4}=\langle\sqrt{-1}\rangle\simeq C_{4}$ be the group of fourth roots of unity, then the character group $G^{\ast}:=\mathrm{Hom}(G,\mu_{4})\simeq G$ of $G$ consists of four characters $\{1=\chi_{0},\chi_{1},\chi_{2},\chi_{3}\}$ such that $\chi_{j}(\tau)=\sqrt{-1}^{j}$ for $0\leq j\leq 3$. The values of these characters for $g\in G$ are shown in Table 7.2.

Four central orthogonal idempotents in the group ring $R=\mathbb{C}[G]$ are given generally by

[TABLE]

*More explicitly, we have *

$\psi_{0}=\frac{1}{4}(1+\tau+\tau^{2}+\tau^{3}),\ \psi_{1}=\frac{1}{4}(1-\sqrt{-1}\tau-\tau^{2}+\sqrt{-1}\tau^{3}),\ \psi_{2}=\frac{1}{4}(1-\tau+\tau^{2}-\tau^{3})$, and $\psi_{3}=\frac{1}{4}(1+\sqrt{-1}\tau-\tau^{2}-\sqrt{-1}\tau^{3})$, and consequently the sum relation $\psi_{0}+\psi_{1}+\psi_{2}+\psi_{3}=1$.

Proof.

For all $0\leq i,j\leq 3$, we have $\psi_{i}\cdot\psi_{j}=$

[TABLE]

since $\overline{\chi_{i}(\tau)}\chi_{j}(\tau)$ is a fixed fourth root of unity $\zeta$ which satisfies $1+\zeta+\zeta^{2}+\zeta^{3}=0$ for $i\neq j$, $\zeta\neq 1$, and $1+\zeta+\zeta^{2}+\zeta^{3}=4$ for $i=j$, $\zeta=1$. (Here, $\delta_{i,j}$ denotes the Kronecker delta.) ∎

For our application to the arithmetic of pure metacyclic fields $N=\mathbb{Q}(\root 5 \of{D},\zeta_{5})$ of degree $20$, we can view the four central orthogonal idempotents as elements of the group ring $(\mathbb{Z}/5\mathbb{Z})[G]$ using the isomorphism $\mu_{4}=\langle\sqrt{-1}\rangle\simeq C_{4}\simeq U(\mathbb{Z}/5\mathbb{Z})$ which arises by mapping $\sqrt{-1}\mapsto\bar{3}$, $-1\mapsto\bar{4}$, $-\sqrt{-1}\mapsto\bar{2}$. Then we obtain:

[TABLE]

7.1. Invariance

Based on the preceding discussion of central orthogonal idempotents, let us now consider the $\tau$-invariance of associated particular ambiguous ideals in the semi-local $\mathbb{F}_{5}$-subspace

[TABLE]

with respect to a prime number $\ell\in\mathbb{P}$ such that $\ell\equiv 1\,(\mathrm{mod}\,5)$, $\ell\mathcal{O}_{L}=\mathfrak{l}^{5}$, $\mathfrak{l}\mathcal{O}_{M}=\mathcal{L}\cdot\mathcal{L}^{\tau}$, and $\mathcal{L}\mathcal{O}_{N}=\mathfrak{L}\cdot\mathfrak{L}^{\tau^{2}}$, $\mathcal{L}^{\tau}\mathcal{O}_{N}=\mathfrak{L}^{\tau}\cdot\mathfrak{L}^{\tau^{3}}$ with $\ell\in\mathbb{P}_{L}$, $\mathcal{L}\in\mathbb{P}_{M}$, and $\mathfrak{L}\in\mathbb{P}_{N}$.

Proposition 7.1**.**

1. (1)

The $4$-norm image element $\mathfrak{l}\mathcal{O}_{N}=\mathfrak{L}^{1+\tau^{2}+\tau+\tau^{3}}\hat{=}(1111)$ is $\tau$-invariant. 2. (2)

The $2$-norm kernel element $\mathcal{K}\mathcal{O}_{N}:=\mathcal{L}\mathcal{O}_{N}(\mathcal{L}^{\tau}\mathcal{O}_{N})^{4}=\mathfrak{L}^{1+\tau^{2}+4\tau+4\tau^{3}}\hat{=}(1414)$ is not $\tau$-invariant but is mapped to its (linearly dependent) inverse $\mathcal{K}^{-1}\mathcal{O}_{N}\hat{=}(4141)$ by $\tau$. 3. (3)

*The $4$-norm kernel element $\mathfrak{K}_{1}:=\mathfrak{L}^{1+4\tau^{2}+2\tau+3\tau^{3}}\hat{=}(1243)$ is not $\tau$-invariant but is mapped to its (linearly dependent) third power $\mathfrak{K}_{1}^{3}\hat{=}(3124)$ by $\tau$. Similarly, *

the $4$-norm kernel element $\mathfrak{K}_{2}:=\mathfrak{L}^{1+4\tau^{2}+3\tau+2\tau^{3}}\hat{=}(1342)$ is not $\tau$-invariant but is mapped to its (linearly dependent) second power $\mathfrak{K}_{2}^{2}\hat{=}(2134)$ by $\tau$. 4. (4)

*The $4$-norm kernel element $\mathfrak{L}^{1+4\tau^{2}+\tau+4\tau^{3}}\hat{=}(1144)$ is not $\tau$-invariant and is mapped to the linearly independent $4$-norm kernel element $(4114)$ by $\tau$. Similarly, *

$\mathfrak{L}^{1+4\tau^{2}}\hat{=}(1040)$, resp. $(0104)$, is mapped to the linearly independent $(0104)$, resp. $(4010)$.

Proof.

1. (1)

$(\mathfrak{L}^{1+\tau^{2}+\tau+\tau^{3}})^{\tau}=\mathfrak{L}^{\tau+\tau^{3}+\tau^{2}+1}\hat{=}(1111)$. 2. (2)

$(\mathfrak{L}^{1+\tau^{2}+4\tau+4\tau^{3}})^{\tau}=\mathfrak{L}^{\tau+\tau^{3}+4\tau^{2}+4}\hat{=}(4141)$. 3. (3)

$(\mathfrak{L}^{1+4\tau^{2}+2\tau+3\tau^{3}})^{\tau}=\mathfrak{L}^{\tau+4\tau^{3}+2\tau^{2}+3}\hat{=}(3124)$, $(\mathfrak{L}^{1+4\tau^{2}+3\tau+2\tau^{3}})^{\tau}=\mathfrak{L}^{\tau+4\tau^{3}+3\tau^{2}+2}\hat{=}(2134)$. 4. (4)

$(\mathfrak{L}^{1+4\tau^{2}+\tau+4\tau^{3}})^{\tau}=\mathfrak{L}^{\tau+4\tau^{3}+\tau^{2}+4}\hat{=}(4114)$, $(\mathfrak{L}^{1+4\tau^{2}})^{\tau}=\mathfrak{L}^{\tau+4\tau^{3}}\hat{=}(0104)$,

$(\mathfrak{L}^{\tau+4\tau^{3})^{\tau}=\mathfrak{L}^{\tau^{2}+4}}\hat{=}(4010).\qed$

Theorem 7.1**.**

For a pure metacyclic field with one of the types $\alpha_{2}$, $\beta_{1}$, $\delta_{1}$, $\zeta_{1}$, the one-dimensional relative principal factorization in the case $s_{4}=1$ must be generated by

[TABLE]

In any case $s_{4}\geq 1$, it cannot be generated by any of the ideals

[TABLE]

Proof.

In the case $s_{4}=1$, the only two $\tau$-invariant $1$-dimensional subspaces of the $2$-dimensional $\mathbb{F}_{5}$-vectorspace of primitive relatively ambiguous ideals $(\mathcal{I}_{N/K}/\mathcal{I}_{K})\bigcap\ker{N_{N/L}}$ are generated by

[TABLE]

according to Proposition 7.1. If, for instance, $\mathfrak{K}_{1}=A\mathcal{O}_{N}$ is principal, then $(3124)\hat{=}\mathfrak{L}^{\tau+4\tau^{3}+2\tau^{2}+3}=(\mathfrak{L}^{1+4\tau^{2}+2\tau+3\tau^{3}})^{\tau}=A^{\tau}\mathcal{O}_{N}$ is also principal, but being equal to the third power $\mathfrak{K}_{1}^{3}$ it also belongs to the $1$-dimensional subspace generated by $\mathfrak{K}_{1}$.

In any case $s_{4}\geq 1$, we give the proof exemplarily for $\mathfrak{L}^{1+4\tau^{2}}$. If $(1040)\hat{=}\mathfrak{L}^{1+4\tau^{2}}=A\mathcal{O}_{N}$ were principal, then $(0104)\hat{=}\mathfrak{L}^{\tau+4\tau^{3}}=(\mathfrak{L}^{1+4\tau^{2}})^{\tau}=A^{\tau}\mathcal{O}_{N}$ would also be principal, according to Proposition 7.1. Since $(1040)$ and $(0104)$ are linearly independent over $\mathbb{F}_{5}$, we would obtain a two-dimensional relative principal factorization, which is impossible for the types $\alpha_{2}$, $\beta_{1}$, $\delta_{1}$, $\zeta_{1}$. (This can occur only for a field with type $\alpha_{1}$.) ∎

7.2. Application to metacyclic fields

For the investigation of differential principal factors, i.e., ambiguous principal ideals, in the relative extensions $L/\mathbb{Q}$, $M/K^{+}$ and $N/K$, it is illuminating to begin by determining the order of the finite group of all primitive ambiguous ideals of these extensions. The word “primitive” always refers to the base field of an extension. Denoting by $G:=\mathrm{Gal}(N/K)$ the subgroup of the metacyclic group $\mathrm{Gal}(N/\mathbb{Q})=\langle\sigma,\tau\rangle$ which is generated by the automorphism $\sigma$ of order $5$, we let $\mathcal{I}_{N}^{G}:=\{\mathfrak{A}\in\mathcal{I}_{N}\mid\mathfrak{A}^{\sigma}=\mathfrak{A}\}$, $\mathcal{I}_{M}^{G}:=\mathcal{I}_{M}\cap\mathcal{I}_{N}^{G}$ and $\mathcal{I}_{L}^{G}:=\mathcal{I}_{L}\cap\mathcal{I}_{N}^{G}$. Note that these groups correspond to $\mathcal{I}_{N/K}=\mathcal{I}_{N}^{G}$, $\mathcal{I}_{M/K^{+}}=\mathcal{I}_{M}^{G}$ and $\mathcal{I}_{L/\mathbb{Q}}=\mathcal{I}_{L}^{G}$.

Theorem 7.2**.**

Let $R$ be the product of all prime numbers ramified in $L$ and put

[TABLE]

1. (1)

The group of absolute primitive ambiguous ideals is of order $5^{T}$,

[TABLE]

and it is characterized uniquely with the aid of absolute norms in $\mathbb{Z}$. 2. (2)

The group of intermediate primitive ambiguous ideals is of order $5^{T+s_{2,4}}$,

[TABLE]

and $(\mathcal{I}_{M}^{G}/\mathcal{I}_{K^{+}})/\ker(N_{M/L})\simeq\mathcal{I}_{L}^{G}/\mathcal{I}_{\mathbb{Q}}$, where the norm kernel of order $5^{s_{2,4}}$ is

[TABLE]

Here it is assumed that $\ell\mathcal{O}_{M}=(\mathcal{L}^{1+\tau})^{5}$ with $\mathcal{L}\in\mathbb{P}_{M}$ for $\ell\equiv\pm 1\,(\mathrm{mod}\,5)$. 3. (3)

The group of relative primitive ambiguous ideals is of order $5^{T+s_{2,4}+2s_{4}}$,

[TABLE]

and $(\mathcal{I}_{N}^{G}/\mathcal{I}_{K})/\ker(N_{N/M})\simeq\mathcal{I}_{M}^{G}/\mathcal{I}_{K^{+}}$, where the norm kernel of order $5^{2s_{4}}$ is

[TABLE]

*Here it is assumed that $\ell\mathcal{O}_{N}=(\mathfrak{L}^{1+\tau^{2}+\tau+\tau^{3}})^{5}$ with $\mathfrak{L}\in\mathbb{P}_{N}$ for $\ell\equiv+1\,(\mathrm{mod}\,5)$. *

Summarized* and expressed as a direct product of elementary abelian $5$-groups:*

[TABLE]

Proof.

Generally, for any prime number $q\in\mathbb{P}$ which divides $R$, that is $v_{q}(R)=1$, we have full ramification $q\mathcal{O}_{L}=\mathfrak{q}^{5}$ in $L$ with a prime ideal $\mathfrak{q}\in\mathbb{P}_{L}$.

1. (1)

Since the extension $L/\mathbb{Q}$ is of prime degree $5$, an ambiguous prime ideal $\mathfrak{q}$ in $\mathcal{I}_{L}^{G}$ is either inert or totally ramified over $\mathbb{Q}$. In the former case it is imprimitive, $\mathfrak{q}\in\mathcal{I}_{\mathbb{Q}}$, and in the latter case it lies over a prime divisor $q$ of $R$. The relation $\mathfrak{q}^{5}=\mathfrak{q}^{1+\sigma+\ldots+\sigma^{4}}=N_{L/\mathbb{Q}}\mathfrak{q}=q\mathbb{Z}\in\mathcal{I}_{\mathbb{Q}}$ shows that a primitive ambiguous ideal must be free of fifth powers, whence $\mathcal{I}_{L}^{G}/\mathcal{I}_{\mathbb{Q}}\simeq\prod_{q\mid R}\,\{\mathfrak{q}^{v_{q}}\mid 0\leq v_{q}\leq 4\}$ and multiplication involves reduction of exponents modulo $5$. If $R=q_{1}\cdots q_{T}$, the norm map $N_{L/\mathbb{Q}}$ establishes an isomorphism $\mathcal{I}_{L}^{G}/\mathcal{I}_{\mathbb{Q}}\simeq N_{L/\mathbb{Q}}(\mathcal{I}_{L}^{G})/(\mathbb{Q}^{\times})^{5}=(\mathbb{Q}^{\times})^{5}\cdot\langle q_{1},\ldots,q_{T}\rangle/(\mathbb{Q}^{\times})^{5}\simeq\langle q_{1},\ldots,q_{T}\rangle/(\langle q_{1},\ldots,q_{T}\rangle\cap(\mathbb{Q}^{\times})^{5})=$

$\langle q_{1},\ldots,q_{T}\rangle/\langle q_{1},\ldots,q_{T}\rangle^{5}\simeq(\langle q_{1}\rangle/\langle q_{1}\rangle^{5})\times\ldots\times(\langle q_{T}\rangle/\langle q_{T}\rangle^{5})\simeq(\mathbb{Z}/5\mathbb{Z})^{T}$. 2. (2)

For the intermediate extension $M/K^{+}$, we have to take into account the decomposition law for the primes $q\equiv\pm 1\,(\mathrm{mod}\,5)$. A similar reasoning as in item (1) shows that $\mathcal{I}_{M}^{G}/\mathcal{I}_{K^{+}}\simeq\mathcal{I}_{1}\times\mathcal{I}_{2}$ is the direct product of two components, $\mathcal{I}_{1}:=\prod_{q\not\equiv\pm 1\,(\mathrm{mod}\,5)}\,\{\mathcal{Q}^{v_{q}}\mid 0\leq v_{q}\leq 4\}$ of order $5^{t-s_{2}}$ and, semi-locally with respect to $q$, $\mathcal{I}_{2}:=\prod_{q\equiv\pm 1\,(\mathrm{mod}\,5)}\,\{\mathcal{Q}^{v_{q}}(\mathcal{Q}^{\tau})^{v_{q}^{\prime}}\mid 0\leq v_{q},v_{q}^{\prime}\leq 4\}$ of order $(5^{2})^{s_{2}}=5^{2s_{2}}$.

The norm map $N_{M/L}$ is injective on $\mathcal{I}_{1}$ but has kernel $\ker(N_{M/L})=\prod_{q\equiv\pm 1(5)}\,\{\mathcal{Q}^{v_{q}}(\mathcal{Q}^{\tau})^{v_{q}^{\prime}}\mid v_{q}+v_{q}^{\prime}\equiv 0\,(\mathrm{mod}\,5)\}$ of order $5^{s_{2}}$ on $\mathcal{I}_{2}$. For the justification of these claims observe, firstly, that $N_{M/L}(\mathcal{Q})=\mathfrak{q}^{2}$, with exponent $2$ coprime to $5$, for $q\not\equiv\pm 1\,(\mathrm{mod}\,5)$, $q\neq 5$, since $\mathfrak{q}$ remains inert in $M$, secondly, that $N_{M/L}(\mathcal{Q})=\mathfrak{q}$ for $q=5$, since $\mathfrak{q}\mathcal{O}_{M}=\mathcal{Q}^{2}$ ramifies in $M$. 3. (3)

Similar as (2). Formula (7.11) corresponds to formulas (1.4) and (6.3).∎

8. Rational congruence conditions and asymptotic limit densities

Using connections between differential principal factorization types and invertible residue classes, we prove that the asymptotic limit density of certain types is zero.

The existence, resp. the lack, of certain prime divisors of the conductor $f$ permits some criteria for the classification of pure quintic fields.

Theorem 8.1**.**

1. (1)

$\zeta\in N_{N/K}(U_{N})$* can occur only when $f$ is divisible by no other primes than $5$ or primes $q_{j}\equiv\pm 1,\pm 7\,(\mathrm{mod}\,5^{2})$.* 2. (2)

Relative DPF of $N/K$ can occur only if some prime $q_{j}\equiv+1\,(\mathrm{mod}\,5)$ divides $f$. 3. (3)

Intermediate DPF of $M/K^{+}$ can occur only if some prime $q_{j}\equiv\pm 1\,(\mathrm{mod}\,5)$ divides $f$. 4. (4)

Absolute DPF of $L/\mathbb{Q}$, distinct from radicals, must exist when no prime $\equiv\pm 1\,(\mathrm{mod}\,5)$ divides $f$ and $f$ has at least one prime factor distinct from $5$ and from primes $\equiv\pm 7\,(\mathrm{mod}\,5^{2})$.

Proof.

1. (1)

The condition for $\zeta\in N_{N/K}(U_{N})$ is due to the properties of the quintic Hilbert symbol (fifth power norm residue symbol) over $K$. 2. (2)

The condition for relative DPF of $N/K$ is a consequence of the decomposition law $(e,f,g)=(1,1,4)$ for primes $\equiv+1\,(\mathrm{mod}\,5)$ in $K$. 3. (3)

The condition for intermediate DPF of $M/K^{+}$ is due to the decomposition types $(e,f,g)=(1,1,2)$, resp. $(1,2,2)$, of primes $\equiv-1\,(\mathrm{mod}\,5)$ in $K^{+}$, resp. $K$, and $(e,f,g)=(1,1,2)$ of primes $\equiv+1\,(\mathrm{mod}\,5)$ in $K^{+}$. ∎

Theorem 8.2**.**

The asymptotic limit density of pure quintic fields $L=\mathbb{Q}(\sqrt[5]{D})$ with $5$th power free radicand $D$ and one of the differential principal factorization types $\zeta_{1}$, $\zeta_{2}$, $\eta$, $\vartheta$, that is, where a primitive fifth root of unity $\zeta_{5}$ is representable as norm of a unit in the Galois closure $N=K\cdot L$ with $K=\mathbb{Q}(\zeta_{5})$, i.e. $(\exists Z\in U_{N})\,\mathrm{Norm}_{N/K}(Z)=\zeta_{5}$, vanishes.

Proof.

A necessary condition for the existence of a unit $Z\in U_{N}$ with $\mathrm{Norm}_{N/K}(Z)=\zeta_{5}$ is that the conductor $f$ associated with the fifth power free radicand $D$ has only prime divisors $p=5$ or $p\equiv\pm 1,\pm 7\,(\mathrm{mod}\,25)$ but no prime divisor $q$ in other invertible residue classes $(\mathrm{mod}\,25)$.

The group of invertible residue classes modulo $25$ is

$U(\mathbb{Z}/25\mathbb{Z})=\{1,2,3,4,6,7,8,9,11,12,13,14,16,17,18,19,21,22,23,24\}$ with $20$ elements.

For an increasing number $n$ of prime factors of the conductor $f$, we have the following probabilities $P(n)$ for a constitution by prime divisors $p\equiv\pm 1,\pm 7\,(\mathrm{mod}\,9)$:

• •

for $n=1$: only $4$, namely $1,7,18,24$, among $20$ residue classes, and thus $P(1)=\frac{4}{20}=\frac{1}{5}$,

• •

for $n=2$: only $4^{2}$ among $20^{2}$ pairs, and thus $P(2)=\frac{4^{2}}{20^{2}}=\frac{1}{5^{2}}$,

• •

generally for $n=t$: $P(t)=\frac{1}{5^{t}}$.

Therefore, the limit probability is given by $\lim_{t\to\infty}\,P(t)=\lim_{t\to\infty}\,\frac{1}{5^{t}}=0$. However, the asymptotic limit density of arbitrary positive integers without prime divisors in an entire residue class, or even from several residue classes, is generally zero, independently of the number $n$. ∎

Remark 8.1**.**

In Theorem 8.2, we made use of Dirichlet’s Theorem on the density of primes which populate invertible residue classes.

9. Computational simplification

By means of the following lemma, we shall obtain a reduction of computational complexity for determining the differential principal factorization type of certain pure quintic number fields $L=\mathbb{Q}(\root 5 \of{D})$.

Lemma 9.1**.**

Let $\eta>1$ be the fundamental unit of the real qudratic number field $K^{+}=\mathbb{Q}(\sqrt{5})$, that is $\eta=\frac{1}{2}(1+\sqrt{5})$.

For the unit norm index $U^{+}:=(U_{K^{+}}:N_{M/K^{+}}(U_{M}))\in\{1,5\}$ the following conditions hold:

[TABLE]

Proof.

Since $-1=(-1)^{5}=N_{M/K^{+}}(-1)$ and thus $-1\in N_{M/K^{+}}(U_{M})$, we have

$U^{+}=1$ $\Longleftrightarrow$ $N_{M/K^{+}}(U_{M})=U_{K^{+}}=\langle-1,\eta\rangle$ $\Longleftrightarrow$ $(\exists\,\upsilon\in U_{M})\,N_{M/K^{+}}(\upsilon)=\eta$.

Necessity of the condition: If $\eta=N_{M/K^{+}}(\upsilon)$ for some $\upsilon\in U_{M}<U_{N}$, then we trivially also have $\eta=\upsilon^{Tr(\sigma)}=N_{N/K}(\upsilon)$, where $\mathrm{Gal}(N/K)=\langle\sigma\rangle$ and $Tr(\sigma)=1+\sigma+\sigma^{2}+\sigma^{3}+\sigma^{4}$.

Sufficiency of the condition: If $\eta=N_{N/K}(H)$ for some $H\in U_{N}$, then the commutativity of the norm diagram

implies $\eta^{2}=N_{K/K^{+}}(\eta)=N_{K/K^{+}}(N_{N/K}(H))=N_{M/K^{+}}(N_{N/M}(H))=N_{M/K^{+}}(\upsilon)$, where $\upsilon:=N_{N/M}(H)\in U_{M}$. Finally, $\eta^{2}$ generates $\langle\eta\rangle$ modulo $\eta^{5}$, since $(\eta^{2})^{3}=\eta^{6}\equiv\eta\,(\mathrm{mod}\,\langle\eta^{5}\rangle)$. Thus, we have $U^{+}=1$. ∎

Theorem 9.1**.**

Let $D=q_{1}^{e_{1}}\cdots q_{s}^{e_{s}}$ be a $5$th power free radicand with $s\geq 1$, distinct primes $q_{1},\ldots,q_{s}$ and exponents $1\leq e_{j}\leq 4$. If

[TABLE]

then the following criteria hold for $L=\mathbb{Q}(\root 5 \of{D})$:

1. (1)

$L$* is of type $\varepsilon$ $\Leftrightarrow$ $(U_{K}:N_{N/K}(U_{N}))=5$.* 2. (2)

$L$* is of type $\gamma$ $\Leftrightarrow$ $(U_{K}:N_{N/K}(U_{N}))=25$.*

Proof.

Firstly, the condition $(\forall\,1\leq j\leq s)\,q_{j}\not\equiv\pm 1\,(\mathrm{mod}\,5)$ implies $I=R=0$ and thus discourages the types $\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2},\delta_{1},\delta_{2},\zeta_{1},\zeta_{2}$ for which either $I\geq 1$ or $R\geq 1$. Moreover, the condition also implies $(\forall\,1\leq j\leq s)\,q_{j}\not\equiv\pm 1\,(\mathrm{mod}\,25)$. Secondly, the other condition $(\exists\,1\leq j\leq s)\,(q_{j}\neq 5\text{ and }q_{j}\not\equiv\pm 7\,(\mathrm{mod}\,25))$, together with $(\forall\,1\leq j\leq s)\,q_{j}\not\equiv\pm 1\,(\mathrm{mod}\,25)$, excludes the types $\vartheta,\eta$ and, once more, the types $\zeta_{1},\zeta_{2}$. Consequently, only the types $\gamma$ and $\varepsilon$ remain as possibilities and can be distinguished by means of the unit norm index $U:=(U_{K}:N_{N/K}(U_{N}))$. ∎

Corollary 9.1**.**

Under the assumptions of Theorem 9.1, the decision between type $\gamma$ and $\varepsilon$ can be reduced from the Galois closure $N$ of degree $20$ to the non-Galois intermediate field $M$ of degree $10$.

1. (1)

$L$* is of type $\varepsilon$ $\Leftrightarrow$ $(U_{K^{+}}:N_{M/K^{+}}(U_{M}))=1$.* 2. (2)

$L$* is of type $\gamma$ $\Leftrightarrow$ $(U_{K^{+}}:N_{M/K^{+}}(U_{M}))=5$.*

Proof.

Since the unique type $\vartheta$ with $U=1$ is impossible, the Lemma 9.1 yields the equivalences $U=5\Longleftrightarrow U^{+}=1$ and $U=25\Longleftrightarrow U^{+}=5$. ∎

10. Computational and theoretical results

At the end of his $1975$ article on class numbers of pure quintic fields, Parry suggested verbatim “In conclusion the author would like to say that he believes a numerical study of pure quintic fields would be most interesting” [27, p. 484]. Of course, it would have been rather difficult to realize Parry’s desire in $1975$. But now, $40$ years later, we are in the position to use the powerful computer algebra systems PARI/GP [26] and MAGMA [5, 6, 17] for starting an attack against this hard problem. This will actually be done in the present paper.

Even in $1991$, when we generalized Barrucand and Cohn’s theory [2] of principal factorization types from pure cubic fields $\mathbb{Q}(\root 3 \of{D})$ to pure quintic fields $\mathbb{Q}(\root 5 \of{D})$ [19], it was still impossible to verify our hypothesis about the distinction between absolute, intermediate and relative differential principal factors (§ 6.3) and about the values of the unit norm index $(U_{K}:N_{N/K}(U_{N}))$ (§ 6.1) by actual computations.

All these conjectures have been proven by our most recent numerical investigations. Our classification is based on the Hasse theorem about the Herbrand quotient of the unit group $U_{N}$ of the Galois closure $N$ as a module over the relative group $G=\mathrm{Gal}(N/K)$ with respect to the cyclotomic subfield $K$. It only involves the unit norm index $(U_{K}:N_{N/K}(U_{N}))$ and our $13$ types of differential principal factors, but not the index of subfield units $(U_{N}:U_{0})$ in Parry’s class number formula (5.1).

10.1. Refined Dedekind species and prototypes

As before, let $N=\mathbb{Q}(\root 5 \of{D},\zeta)$ be a pure metacyclic field of degree $20$ with normalized radicand $D=5^{e_{\ast}}\cdot q_{1}^{e_{1}}\ldots q_{t}^{e_{t}}$ where $0\leq e_{\ast}\leq 4$. The fourth power of the conductor has the shape $f^{4}=5^{e_{0}}\cdot q_{1}^{4}\cdots q_{t}^{4}$ with $e_{0}\in\{0,2,6\}$.

For the primes dividing the conductor, the splitting behavior is described by the number $s_{2}=\#\{1\leq j\leq t\mid q_{j}\equiv-1\,(\mathrm{mod}\,5)\}$ of $2$-split primes in $N$ and the number $s_{4}=\#\{1\leq j\leq t\mid q_{j}\equiv+1\,(\mathrm{mod}\,5)\}$ of $4$-split primes in $N$, and the role in formula [20, Thm. 2, p. 104] for the multiplicity $m$ is described by the number $u=\#\{1\leq j\leq t\mid q_{j}\equiv\pm 1,\pm 7\,(\mathrm{mod}\,25)\}$ of free primes and the number $v=t-u$ of restrictive primes. $n:=t-s_{2}-s_{4}$ is the number of non-split primes.

Definition 10.1**.**

The multiplet $(e_{0};t,u,v,m;n,s_{2},s_{4})$ is called the refined Dedekind species of the field $N$. The first entry characterizes the (coarse) Dedekind species:

[TABLE]

Definition 10.2**.**

A normalized radicand $D$ is called a prototype if it is minimal among all normalized radicands sharing a common

• •

refined Dedekind species $(e_{0};t,u,v,m;n,s_{2},s_{4})$,

• •

differential principal factorization type $(U,\eta,\zeta;A,I,R)$,

• •

set of abelian type invariants of the $5$-class groups $\mathrm{Cl}_{5}(F)$ with $F\in\{L,M,N\}$ and logarithmic unit index $E=v_{5}((U_{N}:U_{0}))$.

Note that, since $t$ is unbounded, there are infinitely many prototypes $D$ of pure metacyclic fields $N$ of degree $20$.

10.2. Prime radicands

Trivially, any prime radicand is fifth power free and normalized, a priori.

Theorem 10.1**.**

Let $L=\mathbb{Q}(\root 5 \of{D})$ be a pure quintic field with prime radicand $D=q\in\mathbb{P}$.

1. (1)

If $q\equiv\pm 2\,(\mathrm{mod}\,5)$ but $q\not\equiv\pm 7\,(\mathrm{mod}\,25)$, then $N$ is a Polya field of type $\varepsilon$. 2. (2)

If $q=5$ or $q\equiv\pm 7\,(\mathrm{mod}\,25)$, then $N$ is a Polya field of type $\vartheta$.

Proof.

In both cases, since $q\not\equiv\pm 1\,(\mathrm{mod}\,5)$, it follows that $q$ does not split in $K^{+},M,K,N$. Consequently, intermediate and relative principal factors are discouraged, $I=R=0$, and only absolute principal factors are possible, $A\geq 1$. This eliminates the possibility of the types $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$, $\beta_{1}$, $\beta_{2}$, $\delta_{1}$, $\delta_{2}$, $\zeta_{1}$, $\zeta_{2}$, and there only remain the types $\gamma$, $\varepsilon$, $\eta$ and $\vartheta$.

1. (1)

Since $q\not\equiv\pm 7\,(\mathrm{mod}\,25)$, the field $L$ is of the species 1b with $f^{4}=5^{2}q^{4}$. Since the additional prime $5$ is ramified in $L$, we have $A\leq 2$, which discourages type $\gamma$ but enables types $\varepsilon$ and $\eta$. However, $q\not\equiv\pm 7\,(\mathrm{mod}\,25)$ together with $q\neq 5$ denies the existence of a unit $Z\in U_{N}$ with $N_{N/K}(Z)=\zeta$, whence types $\eta$ and $\vartheta$ are impossible. So the type must be $\varepsilon$ with invariants $(A,I,R)=(2,0,0)$ and there must exist a unit $H\in U_{N}$ with $N_{N/K}(H)=\eta$. 2. (2)

For $q=5$, the field $L$ is of the species 1a with $f^{4}=5^{6}$. For $q\equiv\pm 7\,(\mathrm{mod}\,25)$, the field $L$ is of the species 2 with $f^{4}=q^{4}$. Since the conductor $f$ is divisible by a single prime only, we have $A\leq 1$, there are only $5^{A}=5$ possible absolute principal factors, and they are occupied by the trivial radicals $\delta^{e}$ with $0\leq e\leq 4$, where $\delta:=\root 5 \of{q}$. Consequently, the invariants $(A,I,R)$ are given by $(1,0,0)$ which uniquely characterizes type $\vartheta$ with $(U_{K}:N_{N/K}(U_{N}))=1$, and enforces the existence of units $Z,H\in U_{N}$ such that $N_{N/K}(Z)=\zeta$ and $N_{N/K}(H)=\eta$. ∎

10.3. Splitting prime radicands

Let $\ell\in\mathbb{P}$ be a prime number which splits in $M$. Suppose the pure quintic field $L=\mathbb{Q}(\sqrt[5]{D})$ is generated by the prime radicand $D=\ell$. Then there arise two situations for the conductor $f$ of $N/K$, either $f^{4}=\ell^{4}$ when $\ell\equiv\pm 1\,(\mathrm{mod}\,25)$ or $f^{4}=5^{2}\cdot\ell^{4}$ when $\ell\equiv\pm 1\,(\mathrm{mod}\,5)$ but $\ell\not\equiv\pm 1\,(\mathrm{mod}\,25)$.

10.4. Prime conductors

1. (1)

$\ell\equiv-1\,(\mathrm{mod}\,25)$:

Example 10.1**.**

There are $6$ occurrences up to the upper bound $D<10^{3}$:

[TABLE]

Here, $N$ is a Polya field and exclusively of type $\delta_{2}$ but never of the types $\zeta_{2}$ or $\vartheta$. 2. (2)

$\ell\equiv+1\,(\mathrm{mod}\,25)$:

Example 10.2**.**

There are $6$ occurrences up to the upper bound $D<10^{3}$:

[TABLE]

The corresponding pure metacyclic field $N$ has the Polya property and is of type

either $\alpha_{1}$, for $D\in\{401,701\}$, even with $\mathrm{Cl}_{5}(L)\simeq C_{5}\times C_{5}$,

or $\alpha_{2}$, for $D\in\{151,251,601\}$,

or $\zeta_{1}$, for $D=101$, but never of the types $\delta_{1}$,$\delta_{2}$, $\zeta_{2}$, $\vartheta$.

10.5. Composite conductors

1. (1)

$\ell\equiv-1\,(\mathrm{mod}\,5)$ but not $\ell\equiv-1\,(\mathrm{mod}\,25)$:

Example 10.3**.**

There are $32$ occurrences up to the upper bound $D<10^{3}$

[TABLE]

[TABLE]

The field $N$ is of type

either $\delta_{2}$, for $D\in\{19,29,59,79,89,109,179,229,239,269,389,409,439,479,509,569,619,$

$659,709,719,739,769,809,839,859,919,929\}$, without Polya property,

or $\beta_{2}$, for $D\in\{139,359,419,829\}$, with Polya property,

or $\varepsilon$, for $D=379$, with Polya property. 2. (2)

$\ell\equiv+1\,(\mathrm{mod}\,5)$ but not $\ell\equiv+1\,(\mathrm{mod}\,25)$:

Example 10.4**.**

There are $33$ occurrences up to the upper bound $D<10^{3}$:

[TABLE]

[TABLE]

The pure metacyclic field $N$ is of type

either $\alpha_{1}$, for $D\in\{31,281,761\}$, without Polya property,

or $\alpha_{2}$, for $D\in\{11,41,61,71,131,181,241,311,331,431,491,541,571,631,661,691,811,$

$821,911,941,971\}$, without Polya property,

or $\beta_{1}$, for $D\in\{191,271,641\}$, with Polya property,

or $\delta_{1}$, for $D\in\{211,421,461,521,881,991\}$, without Polya property.

10.6. Fields of the rare type $\alpha_{3}$ with $2$-dimensional intermediate DPF

Theorem 10.2**.**

For all normalized radicands of the shape $D=q^{e_{0}}\cdot\ell_{1}^{e_{1}}\cdot\ell_{2}^{e_{2}}$ with $0\leq e_{0}\leq 4$, $q\equiv\pm 2\,(\mathrm{mod}\,5)$, $1\leq e_{1},e_{2}\leq 4$, $\ell_{1},\ell_{2}\equiv\pm 1\,(\mathrm{mod}\,5)$, where $L=\mathbb{Q}(\sqrt[5]{D})$ is of DPF type $\alpha_{3}$ with maximal invariant $I=2$, the $2$-dimensional $\mathbb{F}_{5}$-vectorspace $(\mathcal{P}_{M/K^{+}}/\mathcal{P}_{K^{+}})\bigcap\ker(N_{M/L})$ of intermediate ambiguous principal ideals is generated by the kernel ideals $\mathcal{K}^{(\ell_{1})}$ and $\mathcal{K}^{(\ell_{2})}$.

Proof.

In fact, we have the extreme situation that $I=s_{2}+s_{4}=2$ and consequently

[TABLE]

that is, all intermediate ambiguous ideals are principal. ∎

Example 10.5**.**

In the range $2\leq D<1000$ of normalized radicands, there are $8$ cases which are covered by Theorem 10.2. For all of them, the logarithmic index of subfield units takes the value $e=2$. None of the corresponding pure metacyclic fields $N$ can be a Polya field.

1. (1)

$D=319=11\cdot 29$ with $e_{0}=0$ of species 1b, 2. (2)

$D=551=19\cdot 29$ with $e_{0}=0$ of species 2, 3. (3)

$D=589=19\cdot 31$ with $e_{0}=0$ of species 1b, 4. (4)

$D=627=3\cdot 11\cdot 19$ with $e_{0}=1$ of species 1b, 5. (5)

$D=649=11\cdot 59$ with $e_{0}=0$ of species 2, 6. (6)

$D=869=11\cdot 79$ with $e_{0}=0$ of species 1b, 7. (7)

$D=899=29\cdot 31$ with $e_{0}=0$ of species 2, 8. (8)

$D=957=3\cdot 11\cdot 29$ with $e_{0}=1$ of species 2.

Except for $D=627$ where $V_{L}=3$, $V_{M}=4$, $V_{N}=9$, the logarithmic $5$-class numbers are always given by $V_{L}=2$, $V_{M}=2$, $V_{N}=5$.

10.7. Fields of the types $\gamma$, $\varepsilon$, $\eta$, $\vartheta$ without splitting prime divisors of $f$

Theorem 10.3**.**

*(Pure quintic fields with trivial $5$-class group)

Let $L=\mathbb{Q}(\sqrt[5]{D})$ be a pure quintic field, $N=\mathbb{Q}(\zeta_{5},\sqrt[5]{D})$ be the pure metacyclic normal closure of $L$, and $M=\mathbb{Q}(\sqrt{5},\sqrt[5]{D})$ be the maximal real subfield of $N$. For the class numbers $h_{L}$, $h_{M}$ and $h_{N}$ of $L$, $M$ and $N$, the following implications describe divisibility by $5$:*

[TABLE]

Proof.

In terms of $5$-valuations, we prove the contrapositive equivalence $v_{5}(h_{L})=0\Longleftrightarrow v_{5}(h_{M})=0$ using the abbreviations $V_{L}:=v_{5}(h_{L})$ and $V_{M}:=v_{5}(h_{M})$. Let $\mathcal{D}$ be the value of the determinant used by Kobayashi to describe the relation between the unit groups $U_{L}$ of $L$ and $U_{M}$ of $M$. The Kobayashi class number formula $h_{M}=\frac{\mathcal{D}}{5^{2}}h_{L}^{2}$ can be expressed additively by $5$-valuations $V_{M}=2\cdot V_{L}+Q^{+}-2$, where $\mathcal{D}=5^{Q^{+}}$ with $Q^{+}\in\{0,1,2\}$ [13, Prop.2, p.467], [14, Prop.3, p.22].

Firstly, $V_{L}=0$ $\Longrightarrow$ $V_{M}=Q^{+}-2$ and, since $V_{M}$ cannot be negative, $Q^{+}=2$ and $V_{M}=0$.

Secondly, $V_{M}=0$ $\Longrightarrow$ $2\cdot V_{L}=2-Q^{+}\leq 2$ must be even, that is, either $Q^{+}=2$ and $2\cdot V_{L}=0$, whence $V_{L}=0$, or $Q^{+}=0$ and $2\cdot V_{L}=2$, whence $V_{L}=1$. However, the last situation cannot occur, since a class of order $5$ cannot capitulate in $M/L$ of degree $[M:L]=2$.

Furthermore, we prove the contrapositive implications $v_{5}(h_{N})=0\Longrightarrow v_{5}(h_{L})=0$ and $v_{5}(h_{L})=0\Longrightarrow v_{5}(h_{N})\in\{0,1\}$ using the abbreviations $V_{L}:=v_{5}(h_{L})$ and $V_{N}:=v_{5}(h_{N})$. Let $U_{0}$ be the subgroup generated by the units of $K$ and all conjugate fields of $L$ in the unit group $U_{N}$ of $N$. The Parry class number formula $h_{N}=\frac{(U_{N}:U_{0})}{5^{5}}h_{L}^{4}$ can be expressed additively by $5$-valuations $V_{N}=4\cdot V_{L}+Q-5$, where $(U_{N}:U_{0})=5^{Q}$ with $Q\in\{0,\ldots,6\}$ [27, Thm.1, p.476, Thm.2, p.478].

Firstly, $V_{L}=0$ $\Longrightarrow$ $V_{N}=Q-5$ and, since $V_{N}$ cannot be negative, either $Q=5$ and $V_{N}=0$ or $Q=6$ and $V_{N}=1$.

Secondly, $V_{N}=0$ $\Longrightarrow$ $4\cdot V_{L}=5-Q\leq 5$ must be a multiple of $4$, that is, either $Q=5$ and $4\cdot V_{L}=0$, whence $V_{L}=0$, or $Q=1$ and $4\cdot V_{L}=4$, whence $V_{L}=1$. However, the last situation cannot occur, since a class of order $5$ cannot capitulate in $N/L$ of degree $[N:L]=4$. ∎

Corollary 10.1**.**

The relation $5\nmid h_{N}$ enforces an index of subfield units $(U_{N}:U_{0})=5^{5}$ and thus a DPF type either $\varepsilon$ or $\vartheta$. The weaker relation $5\nmid h_{L}$ admits two values of the index of subfield units $(U_{N}:U_{0})\in\{5^{5},5^{6}\}$ and consequently one of the DPF types $\gamma$, $\varepsilon$, $\eta$ or $\vartheta$. Both relations imply the value $\mathcal{D}=5^{2}$ of the Kobayashi determinant.

The cubic analogue of Theorem 10.3 is stated in the following well-known result [11].

Theorem 10.4**.**

*(Pure cubic fields with trivial $3$-class group)

Let $L=\mathbb{Q}(\sqrt[3]{D})$ be a pure cubic field and $N=\mathbb{Q}(\zeta_{3},\sqrt[3]{D})$ be its pure metacyclic normal closure. For the class numbers $h_{L}$ and $h_{N}$ of $L$ and $N$, the following equivalence describes divisibility by $3$:*

[TABLE]

Proof.

In terms of $3$-valuations, we prove the contrapositive equivalence $v_{3}(h_{L})=0\Longleftrightarrow v_{3}(h_{N})=0$ using the abbreviations $V_{L}:=v_{3}(h_{L})$ and $V_{N}:=v_{3}(h_{N})$. Let $U_{0}$ be the subgroup generated by the units of all proper subfields of $N$ in the unit group $U_{N}$ of $N$. The Scholz class number formula $h_{N}=\frac{(U_{N}:U_{0})}{3}h_{L}^{2}$ can be expressed additively by $3$-valuations $V_{N}=2\cdot V_{L}+Q-1$, where $(U_{N}:U_{0})=3^{Q}$ with $Q\in\{0,1\}$ [11, Lem. 1, p. 7].

Firstly, $V_{L}=0$ $\Longrightarrow$ $V_{N}=Q-1$ and, since $V_{N}$ cannot be negative, $Q=1$ and $V_{N}=0$.

Secondly, $V_{N}=0$ $\Longrightarrow$ $2\cdot V_{L}=1-Q\leq 1$ must be even, that is, $Q=1$ and $2\cdot V_{L}=0$, whence $V_{L}=0$. ∎

Corollary 10.2**.**

The relation $3\nmid h_{L}$ enforces an index of subfield units $(U_{N}:U_{0})=3$ and thus a DPF type either $\beta$ or $\gamma$.

10.8. General criterion for Polya fields closely related to DPF types

We begin with a general criterion which is valid for the pure metacyclic normal closures $N=L(\zeta)$ of both, pure cubic fields $L=\mathbb{Q}(\sqrt[3]{D})$ and pure quintic fields $L=\mathbb{Q}(\sqrt[5]{D})$. Let $G=\mathrm{Gal}(N/\mathbb{Q})$ be the absolute Galois group of $N$. Furthermore, denote by $T=\dim_{\mathbb{F}_{5}}(\mathcal{I}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}})$ and by $A=\dim_{\mathbb{F}_{5}}(\mathcal{P}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}})$.

Theorem 10.5**.**

All the following statements are equivalent:

1. (1)

$N$* is a Polya field.* 2. (2)

The subgroup $(\mathcal{I}_{N}^{G}\cdot\mathcal{P}_{N})/\mathcal{P}_{N}\leq\mathrm{Cl}(N)$ of strongly ambiguous classes of $N/\mathbb{Q}$ is trivial. 3. (3)

$\mathcal{I}_{N/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}=\mathcal{P}_{N/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}$* (where $\mathcal{I}_{N/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}\simeq\mathcal{I}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}$ and $\mathcal{P}_{N/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}\simeq\mathcal{P}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}$).* 4. (4)

$\mathcal{I}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}=\mathcal{P}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}$, reduced from the metacyclic normal field to the pure field. 5. (5)

$A=T$, in terms of dimensions over $\mathbb{F}_{5}$ (which necessarily yields an upper bound for the number of primes ramified in $L/\mathbb{Q}$, $T\leq 2$ in the cubic case and $T\leq 3$ in the quintic case). 6. (6)

$(\forall\,p\in\mathbb{P},\ p\mid f)\,(\exists\,\alpha\in L)\,N_{L/\mathbb{Q}}(\alpha)=p$.

Proof.

According to Zantema [37, Thm. 1.3, p. 157], the normal field $N$ with absolute Galois group $G=\mathrm{Gal}(N/\mathbb{Q})$ is a Polya field if and only if the short exact sequence

[TABLE]

collapses to an isomorphism $H^{1}(G,U_{N})\simeq\bigoplus_{p\in\mathbb{P}}\,(\mathbb{Z}/e(p)\mathbb{Z})$, that is, if and only if the Polya group $\mathrm{Po}(N)$, which is generated by the classes of all Ostrowski ideals of $N$ (see formula (1.1)), is trivial. This abstract cohomological statement can be interpreted in the language of algebraic number theory. According to Iwasawa’s isomorphism $H^{1}(G,U_{N})\simeq\mathcal{P}_{N}^{G}/\mathcal{P}_{\mathbb{Q}}=\mathcal{P}_{N/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}$ in formula (1.4), and the isomorphism $\mathcal{I}_{N}^{G}/\mathcal{I}_{\mathbb{Q}}=\mathcal{I}_{N/\mathbb{Q}}/\mathcal{I}_{\mathbb{Q}}\simeq\bigoplus_{p\in\mathbb{P}}\,(\mathbb{Z}/e(p)\mathbb{Z})$ in Corollary 3.2 with $F=\mathbb{Q}$ and $E=N$, the Polya group

[TABLE]

is precisely the group of strongly ambiguous classes of $N/\mathbb{Q}$. Thus, (1) $\Longleftrightarrow$ (2) $\Longleftrightarrow$ (3).

It remains to show that the group of primitive absolutely invariant ideals $\mathcal{I}_{N/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}=\mathcal{I}_{N}^{G}/\mathcal{I}_{\mathbb{Q}}$, which has not appeared in the literature up to now, is isomorphic to the well-known group $\mathcal{I}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}$ in Theorem 4.1. (The group of primitive relatively invariant ideals $\mathcal{I}_{N/K}/\mathcal{P}_{K}=\mathcal{I}_{N}^{H}/\mathcal{I}_{K}$ with the cyclic subgroup $H=\mathrm{Gal}(N/K)$ of the metacyclic group $G$ is also very well known.)

Using the notation of section § 4, and the number $n:=T-s_{2}-s_{4}$ (non-split primes), we have:

[TABLE]

Now we have to select the $\tau$-invariant components of this direct sum, according to section 7.1:

[TABLE]

since $(\mathfrak{L}_{i}^{1-\tau})^{\tau}=(\mathfrak{L}_{i}^{1-\tau})^{-1}$ is not invariant. Therefore, (3) $\Longleftrightarrow$ (4) $\Longleftrightarrow$ (5)$\Longleftrightarrow$ (6). ∎

Finally we apply Theorem 10.5 to a few special situations.

Theorem 10.6**.**

*(The ground state of DPF type $\varepsilon$ with trivial Polya property)

Let $q_{1},q_{2}$ be prime numbers $q_{i}\equiv\pm 2\,(\mathrm{mod}\,5)$ but $q_{i}\not\equiv\pm 7\,(\mathrm{mod}\,25)$. A pure metacyclic field $N=\mathbb{Q}(\zeta,\sqrt[5]{D})$ with conductor $f$ such that*

1. (1)

$f^{4}=5^{2}q_{1}^{4}$* of species $1\mathrm{b}$ forms a singulet, $m=1$, with DPF type $\varepsilon$.* 2. (2)

$f^{4}=5^{6}q_{1}^{4}$* of species $1\mathrm{a}$ belongs to a quartet, $m=4$, with homogeneous DPF type $(\varepsilon,\varepsilon,\varepsilon,\varepsilon)$.* 3. (3)

$f^{4}=q_{1}^{4}q_{2}^{4}$* of species $2$ forms a singulet, $m=1$, with DPF type $\varepsilon$.*

Proof.

We start by proving the multiplicities $m=m(f)$ of the conductors $f$ with the aid of [20, Thm. 2, p. 104], where $t_{0}:=u+v$ denotes the number of primes different from $5$ dividing $f$.

1. (1)

For species $1\mathrm{b}$ with $D=q_{1}^{e_{1}}$, $u=0$, $v=1$, and $D^{4}\not\equiv 1\,(\mathrm{mod}\,25)$, we have $m(f)=(5-1)^{0}\cdot X_{1}=1\cdot\frac{1}{5}(4^{1}-(-1)^{1}))=\frac{5}{5}=1$. 2. (2)

For species $1\mathrm{a}$ with $D=5^{e_{0}}\cdot q_{1}^{e_{1}}$, $t_{0}=1$ and $5\mid D$, we have $m(f)=(5-1)^{t_{0}}=4^{1}=4$. 3. (3)

For species $2$ with $D=q_{1}^{e_{1}}\cdot q_{2}^{e_{2}}$, $u=0$, $v=2$, and $D^{4}\equiv 1\,(\mathrm{mod}\,25)$, we have $m(f)=(5-1)^{0}\cdot X_{2-1}=1\cdot\frac{1}{5}(4^{1}-(-1)^{1}))=\frac{5}{5}=1$.

In each case, we have $T=2$ non-split prime divisors of $f$, $s_{2}=s_{4}=0$, and thus $1\leq A\leq T=2$, by (4.4), $0\leq I\leq s_{2}+s_{4}=0$, by (4.8), $0\leq R\leq 2s_{4}=0$, by (4.12). The assumption $q_{i}\equiv\pm 2\,(\mathrm{mod}\,5)$ but $q_{i}\not\equiv\pm 7\,(\mathrm{mod}\,25)$ excludes the DPF types $\eta$ with $U=1$, $(A,I,R)=(2,0,0)$ and $\vartheta$ with $U=0$, $(A,I,R)=(1,0,0)$, and only the possibility of DPF type $\varepsilon$ with $U=1$, $(A,I,R)=(2,0,0)$, $A=T$ remains. Recall that generally $U+1=A+I+R$, according to (6.1) and (6.3). ∎

Corollary 10.3**.**

In each case, the $5$-class numbers of $L$, $M$, $N$ are given by $h_{L}=h_{M}=h_{N}=5^{w}$ with $w=0$ (ground state of DPF type $\varepsilon$), and the index of subfield units by $5^{E}$ with $E=5$.

Proof.

This claim was proved by Parry in [27, Thm. IV and Formula (10), p. 481]. ∎

Theorem 10.7**.**

*(Ground state of type $\gamma$ with non-trivial Polya property, excited state of type $\varepsilon$)

Let $q_{1},q_{2},q_{3}$ be prime numbers $q_{i}\equiv\pm 2\,(\mathrm{mod}\,5)$ but $q_{i}\not\equiv\pm 7\,(\mathrm{mod}\,25)$. A pure metacyclic field $N=\mathbb{Q}(\zeta,\sqrt[5]{D})$ with conductor $f$ such that*

1. (1)

$f^{4}=5^{2}q_{1}^{4}q_{2}^{4}$* of species $1\mathrm{b}$ forms a triplet, $m=3$, with DPF type $(\gamma^{c},\varepsilon^{b})$, $c+b=3$.* 2. (2)

$f^{4}=5^{6}q_{1}^{4}q_{2}^{4}$* of species $1\mathrm{a}$ belongs to a hexadecuplet, $m=16$, with DPF type $(\gamma^{c},\varepsilon^{b})$, $c+b=16$.* 3. (3)

$f^{4}=q_{1}^{4}q_{2}^{4}q_{3}^{4}$* of species $2$ forms a triplet, $m=3$, with DPF type $(\gamma^{c},\varepsilon^{b})$, $c+b=3$.*

The formal integer exponents $0\leq c,b\leq m$ indicate repetition ($c$ times $\gamma$, $b$ times $\varepsilon$).

Conjecture 10.1**.**

In each case, the $5$-class numbers of $L$, $M$, $N$ are given by either $h_{L}=h_{M}=5^{0}$, $h_{N}=5^{1}$ (ground state of DPF type $\gamma$), and the index of subfield units by $5^{E}$ with $E=6$ or $h_{L}=5^{1}$, $h_{M}=5^{2}$, $h_{N}=5^{4}$ (excited state of DPF type $\varepsilon$), and the index of subfield units by $5^{E}$ with $E=5$.

Proof.

We start by proving the multiplicities $m=m(f)$ of the conductors $f$ with the aid of [20, Thm. 2, p. 104], where $t_{0}:=u+v$ denotes the number of primes different from $5$ dividing $f$.

1. (1)

For species $1\mathrm{b}$ with $D=q_{1}^{e_{1}}\cdot q_{2}^{e_{2}}$, $u=0$, $v=2$, and $D^{4}\not\equiv 1\,(\mathrm{mod}\,25)$, we have $m(f)=(5-1)^{0}\cdot X_{2}=1\cdot\frac{1}{5}(4^{2}-(-1)^{2}))=\frac{15}{5}=3$. 2. (2)

For species $1\mathrm{a}$ with $D=5^{e_{0}}\cdot q_{1}^{e_{1}}\cdot q_{2}^{e_{2}}$, $t_{0}=2$ and $5\mid D$, we have $m(f)=(5-1)^{t_{0}}=4^{2}=16$. 3. (3)

For species $2$ with $D=q_{1}^{e_{1}}\cdot q_{2}^{e_{2}}\cdot q_{3}^{e_{3}}$, $u=0$, $v=3$, and $D^{4}\equiv 1\,(\mathrm{mod}\,25)$, we have $m(f)=(5-1)^{0}\cdot X_{3-1}=1\cdot\frac{1}{5}(4^{2}-(-1)^{2}))=\frac{15}{5}=3$.

In each case, we have $T=3$ non-split prime divisors of $f$, $s_{2}=s_{4}=0$, and thus $1\leq A\leq T=3$, by (4.4), $0\leq I\leq s_{2}+s_{4}=0$, by (4.8), $0\leq R\leq 2s_{4}=0$, by (4.12). The assumption $q_{i}\equiv\pm 2\,(\mathrm{mod}\,5)$ but $q_{i}\not\equiv\pm 7\,(\mathrm{mod}\,25)$ excludes the DPF types $\eta$ with $U=1$, $(A,I,R)=(2,0,0)$ and $\vartheta$ with $U=0$, $(A,I,R)=(1,0,0)$, and only the possibilities of either DPF type $\gamma$ with $U=2$, $(A,I,R)=(3,0,0)$ or DPF type $\varepsilon$ with $U=1$, $(A,I,R)=(2,0,0)$ remain. Recall that generally $U+1=A+I+R$, according to (6.1) and (6.3). Only the fields $N$ of type $\gamma$ have the Polya property $A=T$. ∎

10.9. Numerical tables

The supplementary paper “Tables of pure quintic fields” [22] establishes a complete classification of all $900$ pure metacyclic fields $N=\mathbb{Q}(\zeta,\root 5 \of{D})$ with normalized radicands in the range $2\leq D\leq 1000$. With the aid of PARI/GP [26] and MAGMA [17] we have determined the differential principal factorization type, T, of each field $N$ by means of other invariants $U,A,I,R$. After several weeks of CPU time, the date of completion was Sep. $17$, $2018$.

The possible DPF types are listed in dependence on $U,A,I,R$ in Table 2, where the symbol $\times$ in the column $\eta$, resp. $\zeta$, indicates the existence of a unit $H\in U_{N}$, resp. $Z\in U_{N}$, such that $\eta=N_{N/K}(H)$, resp. $\zeta=N_{N/K}(Z)$. The $5$-valuation of the unit norm index $(U_{K}:N_{N/K}{U_{N}})$ is abbreviated by $U$. The complete statistics is given in Table 3.

The normalized radicand $D=q_{1}^{e_{1}}\cdots q_{s}^{e_{s}}$ of a pure metacyclic field $N$ of degree $20$ is minimal among the powers $D^{n}$, $1\leq n\leq 4$, with corresponding exponents $e_{j}$ reduced modulo $5$. The normalization of the radicands $D$ provides a warranty that all fields are pairwise non-isomorphic.

Prime factors are given for composite $D$ only. Dedekind’s species, S, of radicands is refined by distinguishing $5\mid D$ (species 1a) and $\gcd(5,D)=1$ (species 1b) among radicands $D\not\equiv\pm 1,\pm 7\,(\mathrm{mod}\,25)$ (species 1). By the species and factorization of $D$, the shape of the conductor $f$ is determined. We give the fourth power $f^{4}$ to avoid fractional exponents. Additionally, the multiplicity $m$ indicates the number of non-isomorphic fields sharing a common conductor $f$. The symbol $V_{F}$ briefly denotes the $5$-valuation of the order $h(F)=\#\mathrm{Cl}(F)$ of the class group $\mathrm{Cl}(F)$ of a number field $F$. By $E$ we denote the exponent of the power in the unit index $(U_{N}:U_{0})=5^{E}$.

An asterisk denotes the smallest radicand with given Dedekind species, DPF type and $5$-class groups $\mathrm{Cl}_{5}(F)$, $F\in\{L,M,N\}$. The latter are usually elementary abelian, except for the cases indicated by an additional asterisk.

Principal factors, P, are listed when their constitution is not a consequence of the other information. According to Theorem 7.2, item (1), it suffices to give the rational integer norm of absolute principal factors. For intermediate principal factors, we use the symbols $\mathcal{K}:=\mathcal{L}^{1-\tau}=\alpha\mathcal{O}_{M}$ with $\alpha\in M$ or $\mathcal{L}=\lambda\mathcal{O}_{M}$ with a prime element $\lambda\in M$ (which implies $\mathcal{L}^{\tau}=\lambda^{\tau}\mathcal{O}_{M}$ and thus also $\mathcal{K}=\lambda^{1-\tau}\mathcal{O}_{M}$). Here, $(\mathcal{L}^{1+\tau})^{5}=\ell\mathcal{O}_{M}$ when a prime $\ell\equiv\pm 1\,(\mathrm{mod}\,5)$ divides the radicand $D$. For relative principal factors, we use the symbols $\mathfrak{K}_{1}:=\mathfrak{L}^{1-\tau^{2}+2\tau-2\tau^{3}}=A_{1}\mathcal{O}_{N}$ and $\mathfrak{K}_{2}:=\mathfrak{L}^{1-\tau^{2}-2\tau+2\tau^{3}}=A_{2}\mathcal{O}_{N}$ with $A_{1},A_{2}\in N$. Here, $(\mathfrak{L}^{1+\tau+\tau^{2}+\tau^{3}})^{5}=\ell\mathcal{O}_{N}$ when a prime number $\ell\equiv+1\,(\mathrm{mod}\,5)$ divides the radicand $D$. (Kernel ideals in § 7.)

The quartet $(1,2,4,5)$ indicates conditions which either enforce a reduction of possible DPF types or enable certain DPF types. The lack of a prime divisor $\ell\equiv\pm 1\,(\mathrm{mod}\,5)$ together with the existence of a prime divisor $q\not\equiv\pm 7\,(\mathrm{mod}\,25)$ and $q\neq 5$ of $D$ is indicated by a symbol $\times$ for the component $1$. In these cases, only the two DPF types $\gamma$ and $\varepsilon$ can occur.

A symbol $\times$ for the component $2$ emphasizes a prime divisor $\ell\equiv-1\,(\mathrm{mod}\,5)$ of $D$ and the possibility of intermediate principal factors in $M$, like $\mathcal{L}$ and $\mathcal{K}$. A symbol $\times$ for the component $4$ emphasizes a prime divisor $\ell\equiv+1\,(\mathrm{mod}\,5)$ of $D$ and the possibility of relative principal factors in $N$, like $\mathfrak{K}_{1}$ and $\mathfrak{K}_{2}$. The $\times$ symbol is replaced by $\otimes$ if the facility is used completely, and by $(\times)$ if the facility is only used partially.

If $D$ has only prime divisors $q\equiv\pm 1,\pm 7\,(\mathrm{mod}\,25)$ or $q=5$, a symbol $\times$ is placed in component $5$.

In these cases, $\zeta$ can occur as a norm $N_{N/K}(Z)$ of some unit in $Z\in U_{N}$. If it actually does, the $\times$ is replaced by $\otimes$. Here, we only present the first three tables of [22], Tables 4, 5, and 6.

11. Acknowledgements

We gratefully acknowledge that our research was supported by the Austrian Science Fund (FWF): projects J 0497-PHY and P 26008-N25.