Singular integers and p-class group of cyclotomic fields

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Let p be an irregular prime. Let K=(ζ) be the p-cyclotomic field. From Kummer and class field theory, there exist Galois extensions S/ 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:]=p and an explicit formula for the prime decomposition and ramification of the prime number p in the extensions S/K, M/ 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.