Congruence formulae modulo powers of 2 for class numbers of cyclic quartic fields

Abstract

Let K = \( k(\sqrt \theta ) \) be a real cyclic quartic field, k be its quadratic subfield and \( \tilde K = k(\sqrt { - \theta } ) \) be the corresponding imaginary quartic field. Denote the class numbers of K, k and \( \tilde K \) by h K , h k and {417-3} respectively. Here congruences modulo powers of 2 for h − = h K /h K and \( \tilde h^ - = h_{\tilde K} /h_k \) are obtained via studying the p-adic L-functions of the fields.