Note on some p-invariants of Q(N^{1/p}) using reflection theorem

TL;DR

This paper investigates p-class groups of fields generated by p-th roots of integers using reflection theorems, characterizes 3-rational cases, and explores relations between class groups and ramification groups with computational support.

Contribution

It introduces new relations between p-class groups and p-torsion groups via reflection theorems, and characterizes when certain fields are 3-rational, supported by computational data.

Findings

Characterization of N for which L is 3-rational.

Proven equivalence of triviality of Cl(L) and Cl(M).

Provided computational tables for class groups and ramification groups.

Abstract

Let p > 2 be a prime number and let N be any rational integer. We consider the p-class groups Cl(L), Cl(M) of the fields $L:=Q(N^{1/p})$ and $M:=Q(N^{1/p},\mu_p)$, by comparison with the p-torsion groups T(L) and T(M) of the abelian p-ramification theory, in the framework of the reflection theorem, and obtain relations between the ranks of the isotypic components (Theorem 2.6). For p=3, we characterize the integers N such that L is 3-rational (i.e., T(L)=1), giving the following values: $N=3$; $N=3^d \ell$, $\ell = -1+ 3u$; $N=3^d \ell$, $\ell=(1+3a)^2+27b^2$, with $\ell$ prime and $uab$ prime to 3 (Theorem 2.18). We show that the 3-class group Cl(L) is trivial if and only Cl(M) is trivial (Theorem 2.19). We give various tables with PARI/GP programs computing the structure of Cl(L), Cl(M), T(L), T(M), and of the logarithmic class groups (Appendix A, B, C).