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

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Let K = k(√θ) be a real cyclic quartic field, k be its quadratic subfield and K = k(√θ) be the corresponding imaginary quartic field. Denote the class numbers of K, k and K by hK, hk and hK respectively. Here congruences modulo powers of 2 for h = hK /hk
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