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

2011/01/23 by Quême, Roland
#11R18 #11R29 #11R32 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1101.4380

Abstract

Let p be an irregular prime and K=\Q(ζ) the p-cyclotomic field. Let σ 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∈ K\backslash K^p, A\Z_K=\mk a^p with \mk a non-principal ideal of \Z_K, A^σ-μ∈ K^p and μ∈\bf F_p. We compute explicitly the decomposition of the prime p in the subfields M of S of degree [M:\Q]=p.

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