Ten formulae of type Ankeny-Artin-Chowla for class numbers of general cyclic quartic fields (Q1118647)

Let \(p\equiv 1\) (mod 8) be a prime and let \(p=r^ 2+s^ 2\) with s even. Let K be the real cyclic quartic field \({\mathbb{Q}}(\sqrt{p+s\sqrt{p}})\) and k its unique quadratic subfield \({\mathbb{Q}}(\sqrt{p})\). Let h(K) (resp. h(k)) denote the class number of the field K (resp. k). Using the theory of p-adic L-functions the author proves the congruence \(C\cdot h(K)/h(k)\equiv B_{(p-1)/4}B_{3(p-1)/4}\quad (mod\quad p),\) where C is an explicitly given constant and \(B_ n\) denotes the n-th Bernoulli number, as well as a number of similar congruences.