1) Abelian extension K/Q: Let K/Q be an abelian extension with n = [K:Q] > 1, n odd. Suppose that K is not principal. Suppose that p is an odd prime dividing the class number h(K) of K, and such that p-1 has at least one odd prime divisor. Then p \times (p-1) and n are not coprime.

2) Galois extension K/Q: Let K/Q be a Galois extension with n = [K:Q] > 1. Suppose that K is not principal. Let g be an odd prime with g^alpha || n. There exists a cyclic extension K/F with beta \in N, 1 \leq beta \leq alpha and [K:F] = g^beta, such an extension K/F_1 not existing for alpha \geq beta_1 > beta.

Suppose now that p \not= g is an odd prime dividing the class number h(K) of K and not dividing the class number h(F) of F. Let r_p be the rank of the p-class group of K. Then p \equiv 1 \modu g. Suppose moreover that, for all fields E \not= K, F \subset E \subset K, the prime p does not divide the class number h(E). If g \not| r_p then p \equiv 1 \modu g^beta.

Observe that, in that case, this result completes "Rank Theorem" (see Masley \cite{mas} cor 2.15 p 305).

The paper is at elementary level and contains a lot of numerical examples.

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Roland Queme <roland.queme@free.fr>