Infinite class field towers of number fields of prime power discriminant

TL;DR

This paper proves that for every prime p, there exists a solvable number field with ramification only at p and infinity, whose p-Hilbert Class field tower is infinite, advancing understanding of class field towers.

Contribution

It demonstrates the existence of infinite class field towers for specific solvable number fields ramified only at a prime p and infinity.

Findings

Existence of such number fields for all primes p.

Infinite p-Hilbert Class field towers established.

Solvability of the fields involved.

Abstract

For every prime number p, we show the existence of a solvable number field L ramified only at {p and infinity whose p-Hilbert Class field tower is infinite.

Equations

$${\mathrm{V}}_S:=\{x\in {\mathrm{K}}^{\times }\mid x{\mathcal{O}}_{\mathrm{K}}={\mathfrak{A}}^p,x\in {\mathrm{K}}_v^{\times p},\forall v\in S\}.$$

$$\dim H^1({\mathrm{G}}_S,{\mathbb{F}}_p)=\frac{efg}{2}+1+\dim H^2({\mathrm{G}}_S,{\mathbb{F}}_p)$$

$$\dim H^2({\mathrm{G}}_S,{\mathbb{F}}_p)=g-1+\dim {\mathrm{V}}_S/{\mathrm{K}}^{\times p}.$$

$${\omega }_{\mathrm{G}}(g)=\max \{\omega (y),\phi (y)=g\}.$$

$$1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$4\left(g-1+g\frac{(ef+2)(ef+1)}{2}\right)\stackrel{?}{<}\frac{g^2}{4}(ef+2{)}^2$$

$$16(g-1)+8g(ef+2)(ef+1)\stackrel{?}{<}{g}^2(ef+2{)}^2.$$

$$16+8(x+2)(x+1)\stackrel{?}{<}g(x+2{)}^2.$$

Full text

For a number field $\rm L$ of degree $n$ over ${\mathbb{Q}}$, the root discriminant is defined to be $D_{\rm L}^{1/n}$ where $D_{\rm L}$ is the absolute value of the discriminant of $\rm L$. Given a finite set $S$ of places of ${\mathbb{Q}}$, it is an old question as to whether there is an infinite sequence of number fields unramified outside $S$ with bounded root discriminant. This question is related to the constants of Martinet [8] and Odlyzko’s bounds [10]. Since the root discriminant is constant in unramified extensions, an approach to answering the previous question in the positive is to find a number field $\rm L$ (of finite degree) unramified outside $S$ having an infinite class field tower. In the case of ${\rm K}/{\mathbb{Q}}$ quadratic, it is a classical result of Golod and Shafarevich that if ${\rm K}/{\mathbb{Q}}$ is ramified at at least $8$ places, then ${\rm K}$ has an infinite class field tower. On the other hand, if $p$ is a prime, and $S=\{p,\infty\}$, the question becomes whether there exist number fields with $p$-power discriminant having an infinite unramified extension. Schmitals [11] and Schoof [12] produced a few isolated examples of this type. See also [3], [7], etc. For $p\in\{2,3,5\}$, Hoelscher [4] announced the existence of number fields unramified outside $\{p,\infty\}$ and having an infinite Hilbert class field tower. Here we prove:

Our proof is based on the idea of cutting of wild towers introduced in [2]; in particular it does not involve the usual technique of genus theory. For the more refined question where $S$ consists of a single prime number $p$ (i.e. if we focus our attention on totally real fields only), we do not know whether for every prime $p$, there is a totally real number field of $p$-power discriminant having an infinite Hilbert class field tower. In [12, Corollary 4.4] it is shown that ${\mathbb{Q}}(\sqrt{39345017})$ (which is ramified only at the prime $39345017$) has infinite Hilbert class field tower. In [13], Shanks studied primes of the form $p=a^{2}+3a+9$ and the corresponding totally real cubic subfields ${\rm K}\subset{\mathbb{Q}}(\mu_{p})$ and showed the minimal polynomials of ${\rm K}$ are $x^{3}-ax^{2}-(a+3)x-1$. Taking $a=17279$ so $p=298615687$, one can compute that the $2$-part of the class group of ${\rm K}$ has rank $6$. It is not hard to see, using the Golod-Shafarevich criterion, that ${\rm K}$ has infinite $2$-Hilbert class field tower. Thus some examples exist in the totally real case.

1. The results we need

Let $p$ be a prime number. Let ${\rm K}/{\mathbb{Q}}$ be a finite Galois extension. Assume $\mu_{p}\subset{\rm K}$ and moreover that ${\rm K}$ is totally imaginary when $p=2$. For a prime ${\mathfrak{p}}$ of ${\rm K}$ dividing $p$ denote by $e$ (resp. $f$) the ramification index (resp. the residue degree) of ${\mathfrak{p}}$ in ${\rm K}/{\mathbb{Q}}$.

1.1. On the group ${\rm G}_{S}$

Denote by $S$ the set of places of ${\rm K}$ above $p$, and consider ${\rm K}_{S}$ the maximal pro-$p$ extension of ${\rm K}$ unramified outside $S$; put ${\rm G}_{S}={\rm Gal}({\rm K}_{S}/{\rm K})$. Let $g=|S|$ be the number of places of ${\rm K}$ above $p$.

Let $h_{\rm K}^{\prime}$ be the $S$-class number of ${\rm K}$. By class field theory, $h_{\rm K}^{\prime}$ is equal to $[{\rm K}^{\prime}:{\rm K}]$ where ${\rm K}^{\prime}/{\rm K}$ is the maximal abelian of ${\rm K}$ unramified everywhere in which all places of $S$ split completely. The Kummer radical of the $p$-elementary subextension ${\rm K}^{\prime}(p)/{\rm K}$ of ${\rm K}^{\prime}/{\rm K}$ is

[TABLE]

In particular $p\nmid h_{\rm K}^{\prime}$ if and only if ${\rm V}_{S}/{\rm K}^{\times p}$ is trivial.

By work of Koch and Shafarevich the pro-$p$ group ${\rm G}_{S}$ is finitely presented. More precisely, in our situation one has:

Theorem**.**

Let ${\rm K}/{\mathbb{Q}}$ be a totally imaginary Galois extension containing $\mu_{p}$. Let $S=\{p,\infty\}$. Then

[TABLE]

and

[TABLE]

Démonstration.

This is well-known, see for example [9, Corollary 8.7.5 and Theorem 10.7.3]. ∎

We immediately have:

\coroname** \the\smf@thm.**

If $p\nmid h_{\rm K}$ then $\dim H^{1}({\rm G}_{S},{\mathbb{F}}_{p})=g(\frac{ef}{2}+1)$ and $\dim H^{2}({\rm G}_{S},{\mathbb{F}}_{p})=g-1$.

1.2. The cutting towers strategy

1.2.1. The Golod-Shafarevich Theorem

Let ${\rm G}$ be a finitely generated pro-$p$ group. Consider a minimal presentation $1\rightarrow{\rm R}\rightarrow{\rm F}\stackrel{{\scriptstyle\varphi}}{{\rightarrow}}{\rm G}$ of ${\rm G}$, where ${\rm F}$ is a free pro-$p$ group. Set $d=d({\rm G})=d({\rm F})$, the number of generators of ${\rm G}$ and ${\rm F}$. Suppose that ${\rm R}=\langle\rho_{1},\cdots,\rho_{r}\rangle^{Norm}$ is generated as normal subgroup of ${\rm F}$ by a finite set of relations $\rho_{i}$.

We recall the depth function $\omega$ on ${\rm F}$. See [6, Appendice] or [5] for more details. The augmentation ideal $I$ of ${\mathbb{F}}_{p}[[{\rm G}]]$ is, by definition, generated by the set of elements $\{g-e\}_{g\in{\rm G}}$. Then for $e\neq g\in{\rm F}$, define $\omega(g)=\max_{k}\{g-e\in I^{k}\}$; put $\omega(e)=\infty$. It is not difficult to see that $\omega([g,g^{\prime}])\geq 2$ and that $\omega(g^{p^{k}})\geq p^{k}$ for every $g,g^{\prime}\in{\rm G}$ and $k\in{\mathbb{Z}}_{>0}$. Observe also that as the presentation $\varphi$ is minimal, $\omega(\rho_{i})\geq 2$ for all the relations $\rho_{i}$.

The Golod-Shafarevich polynomial associated to the presentation $\varphi$ of ${\rm G}$ is the polynomial $P_{\rm G}(t)=1-dt+\sum_{i}t^{\omega(\rho_{i})}$.

Theorem** (Golod-Shafarevich, Vinberg [14]).**

If ${\rm G}$ is finite then $P_{\rm G}(t)>0$ for all $t\in]0,1[$.

Of course if we have no information about the $\rho_{i}$’s we may take $1-dt+rt^{2}$ (where $r=\dim H^{2}(G,{\mathbb{F}}_{p})$) as Golod-Shafarevich polynomial for ${\rm G}$: if $1-dt+rt^{2}$ is negative at $t_{0}\in]0,1[$, then $P_{\rm G}(t_{0})<0$ and ${\rm G}$ is infinite.

We can also define a depth function $\omega_{\rm G}$ on ${\rm G}$ associated to its augmentation ideal. Then:

\propname** \the\smf@thm.**

For every $g\in{\rm G}$, one has

[TABLE]

Démonstration.

See [6, Appendice 3, Theorem 3.5]. ∎

We now study quotients $\Gamma$ of ${\rm G}$ such that $d({\rm G})=d(\Gamma)$. In this case, the initial minimal presentation of ${\rm G}$ induces a minimal presentation of $\Gamma$

[TABLE]

Suppose that $\Gamma={\rm G}/\langle x_{1},\cdots,x_{m}\rangle^{Norm}$. Here $\langle x_{1},\cdots,x_{m}\rangle^{Norm}$ is the normal subgroup of ${\rm G}$ generated by the $x_{i}$’s. Lift the $x_{i}$’s to $y_{i}\in{\rm F}$ such that $\omega_{\rm G}(x_{i})=\omega(y_{i})$ for each $i$. Hence, $\Gamma={\rm F}/{\rm R}^{\prime}$, where ${\rm R}^{\prime}={\rm R}\langle y_{1},\cdots,y_{m}\rangle^{Norm}$. In particular, if ${\rm R}=\langle\rho_{1},\cdots,\rho_{r}\rangle^{Norm}$, then ${\rm R}^{\prime}=\langle\rho_{1},\cdots,\rho_{r},y_{1},\cdots,y_{m}\rangle^{Norm}$.

If we have no information about the $\rho_{i}$’s, we can take $P_{\Gamma}(t)=1-dt+rt^{2}+\sum_{i}t^{\omega(y_{i})}$ as Golod-Shafarevich polynomial for $\Gamma$.

1.2.2. Cutting of ${\rm G}_{S}$

We want to consider some special quotients $\Gamma$ of ${\rm G}_{S}$, this is what we call “cutting wild towers”.

Each place $v\in S$ corresponds to some extension ${\rm K}_{v}/{\mathbb{Q}}_{p}$ (in fact these fields are isomorphic as ${\rm K}/{\mathbb{Q}}$ is Galois) of degree $ef$. Then, as $\mu_{p}\subset{\rm K}_{v}$, the ${\mathbb{F}}_{p}$-vector space ${\rm K}^{\times}_{v}/{\rm K}^{\times p}_{v}$ has dimension $ef+2$, and local class field theory implies the Galois group of the maximal pro-$p$ extension of ${\rm K}_{v}$ is generated by $ef+2$ elements. Thus the decomposition subgroup ${\rm G}_{v}$ of $v$ in ${\rm K}_{S}/{\rm K}$ is generated by at most $ef+2$ elements $z_{i,v}$. Consider now the commutators $[z_{i,v},z_{k,v}]$ of all these elements; there are at most $\binom{ef+2}{2}$ such elements. Now we cut ${\rm G}_{S}$ by $\langle[z_{i,v},z_{k,v}],i,k;v\in S\rangle^{Norm}$, and denote by $\Gamma$ the corresponding quotient. As $\omega_{{\rm G}_{S}}([z_{i,v},z_{k,v}])\geq 2$, one can take $P_{\Gamma}=1-dt+rt^{2}+g\binom{ef+2}{2}t^{2}$ as Golod-Shafarevich polynomial for $\Gamma$; here $d=\dim H^{1}({\rm G}_{S},{\mathbb{F}}_{p})$ and $r=\dim H^{2}({\rm G}_{S},{\mathbb{F}}_{p})$. This quotient $\Gamma$ of ${\rm G}_{S}$ corresponds to the maximal subextension ${\rm K}_{S}^{loc-ab}/{\rm K}$ of ${\rm K}_{S}/{\rm K}$ locally abelian everywhere. Observe that ${\rm K}_{S}^{loc-ab}/{\rm K}$ contains the compositum of all ${\mathbb{Z}}_{p}$-extensions.

Suppose that there exists some $t_{0}\in]0,1[$ such that $P_{\Gamma}(t_{0})<0$. We will then cut the infinite pro-$p$ group $\Gamma$ by all the $z_{v,i}^{p^{k}}$ for some large $k$. There are $g(ef+2)$ such elements. Denote by $\Gamma_{k}$ the new quotient and by ${\rm K}_{S}^{[k]}$ the new extension of ${\rm K}$ corresponding to $\Gamma_{k}$. Since $\omega_{\Gamma}(z_{v,i}^{p^{k}})\geq p^{k}$, we may take $P_{\Gamma_{k}}(t)=P_{\Gamma}(t)+g(ef+2)t^{p^{k}}$ as the Golod-Shafarevich polynomial for $\Gamma_{k}$. When $k$ is sufficiently large, clearly $P_{\Gamma}(t_{0})<0\implies P_{\Gamma_{k}}(t_{0})<0$, so ${\rm K}_{S}^{[k]}/{\rm K}$ is infinite.

The main interest of ${\rm K}_{S}^{[k]}/{\rm K}$ is:

\propname** \the\smf@thm.**

Suppose ${\rm K}_{S}^{[k]}/{\rm K}$ infinite. Then there exists a finite subextension $\rm L/{\rm K}$ of ${\rm K}_{S}^{[k]}/{\rm K}$ having an infinite Hilbert $p$-class field tower.

Démonstration.

In ${\rm K}_{S}^{[k]}/{\rm K}$ the (wild) ramification is finite: indeed for each $v\in S$, the decomposition groups in ${\rm K}_{S}^{[k]}/{\rm K}$ are abelian, finitely generated and of finite exponent. There exists a finite extension $\rm L/{\rm K}$ inside ${\rm K}_{S}^{[k]}/{\rm K}$ absorbing all the ramification, so ${\rm K}_{S}^{[k]}/\rm L$ is unramified everywhere and infinite. ∎

2. Proof

\propname** \the\smf@thm.**

Let ${\rm K}/{\mathbb{Q}}$ be finite Galois with $\mu_{p}\subset{\rm K}$. Assume that $g\geq 8$.Then there exists a finite subextension $\rm L/{\rm K}$ of ${\rm K}_{S}/{\rm K}$ such that the Hilbert $p$-class field tower of $\rm L$ is infinite.

Démonstration.

Let ${\rm H}$ be the “top” of the Hilbert Class Field Tower of ${\rm K}$. If ${\rm H}/{\rm K}$ is infinite, we are done, so suppose $[{\rm H}:{\rm K}]<\infty$. Note that ${\rm H}$ has class number $1$ so by Corollary 1.1, working over ${\rm H}$, $\dim H^{1}({\rm G}_{S},{\mathbb{F}}_{p})=g\left(\frac{ef}{2}+1\right)$ and $\dim H^{1}({\rm G}_{S},{\mathbb{F}}_{p})=g-1$. As in Section 1.2.2, consider the quotient $\Gamma$ of ${\rm G}_{S}$ by the normal subgroup generated by the local commutators at each $v\in S$; one has $\binom{ef+2}{2}$ such commutators. The group $\Gamma$ can be described by $d:=g\left(\frac{ef}{2}+1\right)$ generators and by $r:=g-1+g\frac{(ef+2)(ef+1)}{2}$ relations.

The Golod-Shafarevich polynomial of $\Gamma$ may be written as $P_{\Gamma}(t)=1-dt+rt^{2}$, when assuming the worst case that all the relations are of depth $2$. Clearly $d/2r<1$, and $P_{\Gamma}(d/2r)=1-\frac{d^{2}}{4r}$. In particular, if $P_{\Gamma}(d/2r)<0$, then one has room to cut by some large $p$-power of the local generators, in order to obtain at the end some finite local groups. For the result to follow, we thus need $4r<d^{2}$, or equivalently

[TABLE]

which is equivalent to

[TABLE]

Replacing the $16(g-1)$ term on the left by $16g$ and dividing by $g$, and setting $x=ef$, we need to verify

[TABLE]

This holds for $g\geq 8$ and $x=ef\geq 1$. Proposition 1.2.2 allows us to conclude ${\rm K}_{S}^{[k]}/{\rm K}$ is infinite when $k$ is sufficiently large. ∎

*Proof Theorem Theorem *: Recall that the principal prime ${\mathfrak{p}}=(1-\zeta_{p^{s}})$ of ${\mathbb{Q}}(\zeta_{p^{s}})$ is the unique prime dividing $p$ and by class field theory ${\mathfrak{p}}$ splits completely in the Hilbert class field ${\rm H}$ of ${\mathbb{Q}}(\zeta_{p^{s}})$. Thus if the class group has order at least $8$, Proposition 2 applied to the solvable number field ${\rm H}$ gives the result.

In the proof of [15, Corollary 11.17], the class number of ${\mathbb{Q}}(\zeta_{p^{r}})$ is shown to be at least $10^{9}$ for $\phi(p^{r})=p^{r-1}(p-1)>220$. Choosing $r\geq 9$ for any $p$ completes the proof of the Theorem.

A slightly more detailed analysis using Table §3 of [15] shows the fields below suffice:

\begin{array}[]{cll}p&{\rm K}&g=h\\ p>23&{\rm K}={\mathbb{Q}}(\zeta_{p})&\geq 8\\ 7\leq p\leq 23&{\rm K}={\mathbb{Q}}(\zeta_{p^{2}})&\geq 43\\ p=5&{\rm K}={\mathbb{Q}}(\zeta_{125})&57708445601\\ p=3&{\rm K}={\mathbb{Q}}(\zeta_{81})&2593\\ p=2&{\rm K}={\mathbb{Q}}(\zeta_{64})&17\end{array}

$\square$

\remaname* \the\smf@thm.*

In [4] a proof of the Theorem for $p=2,3$ and $5$ was given. Our proof is partially modeled on the ideas there, namely considering the Hilbert class field of a cyclotomic field. There are two cases in [4]: Case I, where the Hilbert class field tower is infinite; and Case II, where ramification is allowed at one prime above $p$ in the Hilbert class field ${\rm H}$ and a ${\mathbb{Z}}/p$-extension of ${\rm H}$ ramified at exactly this prime is used. Gras has given a criterion for such an extension to exist: see [1, Chapter V, Corollary 2.4.4]. Gras’ criterion is not verified in [4]. Given the size of the number fields ${\rm H}$, it seems very difficult to do so. We therefore we regard the results of [4] as incomplete.

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