Congruences for the class numbers of real cyclic sextic number fields

Let K6 be a real cyclic sextic number field, and K2, K3 its quadratic and cubic subfield. Let h（L） denote the ideal class number of field L. Seven congruences for h- = h（K6）/（h（K2）h（K3）） are obtained. In particular, when the conductor f6 of K6 is a prime p, （mod p）, where C is an explicitly given constant, and Bn is the Bernoulli number. These results on real cyclic sextic fields are an extension of the results on quadratic and cyclic quartic fields.

LetK 6 be a real cyclic sextic number field, andK 2,K 3 its quadratic and cubic subfield. Leth(L) denote the ideal class number of fieldL. Seven congruences forh - =h (K 6)/(h(K 2)h(K 3)) are obtained. In particular, when the conductorf 6 ofK 6 is a primep, $Ch^ - \equiv B\tfrac{{p - 1}}{6}B\tfrac{{5(p - 1)}}{6}(\bmod p)$ , whereC is an explicitly given constant, andB n is the Bernoulli number. These results on real cyclic sextic fields are an extension of the results on quadratic and cyclic quartic fields.