The structure of some 2-groups and the capitulation problem for certain biquadratic fields
نویسندگان
چکیده
We study the capitulation problem for certain number fields K of degree 4 and we show how we can determine the structures of some 2-groups as an application of this study. Let K (2) 1 be the Hilbert 2-class field of K, K (2) 2 be the Hilbert 2-class field of K (2) 1 , CK,2 be the 2component of the ideal class group of K and G2 the Galois group of K (2) 2 /K. We suppose that CK,2 is of type (2, 2); then K (2) 1 contains three extensions Fi/K, i = 1, 2, 3. The aim of this paper is to study the capitulation of the 2-ideal classes in Fi, i = 1, 2, 3, and as an application of this study, to determine the structure of G2 and the structure of the 2-class group of the three fields Fi, i = 1, 2, 3, for the following cases: (I) K = Q( √ 2q1q2, i) where q1 and q2 are primes such that q1 ≡ q2 ≡ −1 mod 4. This is a case of biquadratic bicyclic number fields of Q. (II) K = Q( √ −pq(2 + √ 2)) where p and q are primes such that p ≡ −q ≡ 5 mod 8. This is a case of quartic number fields of Q.
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