The Hilbert's class field and the p-class group of the cyclotomic fields

TL;DR

This paper investigates the structure of the p-class group and Hilbert's class field of cyclotomic fields, providing explicit computations of prime decomposition in certain unramified extensions related to irregular primes.

Contribution

It offers explicit descriptions of the decomposition of primes in subfields of cyclic unramified extensions of cyclotomic fields, advancing understanding of class groups for irregular primes.

Findings

Explicit prime decomposition in subfields of unramified extensions

Characterization of the p-class group structure

Construction of specific cyclic unramified extensions

Abstract

Let $p$ be an irregular prime and $K=\Q(\zeta)$ the $p$-cyclotomic field. Let $\sigma$ be a $\Q$-isomorphism of $K$ generating $Gal(K/\Q)$. Let $S/K$ be a cyclic unramified extension of degree $p$, defined by $S= K(A^{1/p})$ where $A\in K\backslash K^p$, $A\Z_K=\mk a^p$ with $\mk a$ non-principal ideal of $\Z_K$, $A^{\sigma-\mu}\in K^p$ and $\mu\in{\bf F}_p$. We compute explicitly the decomposition of the prime $p$ in the subfields $M$ of $S$ of degree $[M:\Q]=p$.