Singular integers and p-class group of cyclotomic fields

2006/09/14 by Queme, Roland
#11R18 #11R29 #11R32 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.math/0609410

Abstract

Let p be an irregular prime. Let K=\Q(ζ) be the p-cyclotomic field. From Kummer and class field theory, there exist Galois extensions S/\Q of degree p(p-1) such that S/K is a cyclic unramified extension of degree [S:K]=p. We give an algebraic construction of the subfields M of S with degree [M:\Q]=p and an explicit formula for the prime decomposition and ramification of the prime number p in the extensions S/K, M/\Q and S/M. In the last section, we examine the consequences of these results for the Vandiver's conjecture. This article is at elementary level on Classical Algebraic Number Theory.

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