Formulae of type Ankeny-Artin-Chowla for class numbers of general cyclic sextic fields

Let K 6 be a real cyclic sextic number fields, and K 2, K 3 be its quadratic and cubic subfields. Let h(L) denote the ideal class number of field L. Seven congruences for h -=h(K 6)/h(K 2)h(K 3) are obtained. In particular, when conductor f\-6 of K 6 is prime p, then Ch -≡B p-16B 5(p-1)6 (mod p), where C is an explicitly given constant, and B n is the Bernoulli number. These results for real cyclic sextic fields are an extension of results for quadratic and cyclic quartic fields obtained by Ankeny_Artin_Chowla, Kiselev, Carlitz, Lu Hongwen, Zhang Xianke from 1948 to 1988.

LetK 6 be a real cyclic sextic number fields, andK 2, K3 be its quadratic and cubic subfields. Leth(L) denote the ideal class number of fieldL. Seven congruences forh =h(K6)/ h(K2)h(K3) are obtained. In particular, when conductorf 6 ofK 6 is primep, then whereC is an explicitly given constant, andB n is the Bernoulli number. These results for real cyclic sextic fields are an extension of results for quadratic and cyclic quartic fields obtained by Ankeny-Artin-Chowla. Kiselev. Carlitz. Lu Hongwen, Zhang Xianke from 1948 to 1988.