Let K/Q be a cyclic extension. In this paper, we give several congruences connecting the prime divisors of the degree g= [K:Q] with the prime divisors of the class number h of K/Q. As an exemple, the theorem :

Let K/Q be a cyclic extension with [K:Q]=g. Suppose that g is not divisible by 2 .

Let g_j, j=1,...m, be the prime divisors of g.

Let h_i, i=1,...r, be the prime divisors of the class number h of K/Q.

If for one prime factor h_i of h, the h_i-component G(h_i) of the class group G of K/Q is cyclic then :

- else h_i divides g,
- else h_i = 1 (mod g_j) for at least one prime divisor g_j of g.

The results obtained are all in accordance with class number tables of Washington, Masley, Girtsmair, Schoof, Jeannin and number fields server megrez.math.u-bordeaux.fr

The proofs are strictly elementary.

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Roland Queme <106104.1447@compuserve.com>