On the Isomorphisms of the Galois Groups of the Maximal Abelian Extensions of Imaginary Quadratic Fields

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概要

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• Let Q be the rational number field. For any algebraic number field k of finite degree over Q, we shall denote by A_k the maximal abelian extension of k and by Gal(A_k/k) the Galois group of A_k over k equipped with the Krull topology. The present paper exhibits some counterexamples to the following statement ; for two algebraic number fields k and k' of finite degree over Q, an isomorphism Gal(A_k/k)≅Gal(A_k'/k') of the Galois groups of maximal abelian extensions A_k/k and A_k'/k' implies an isomorphism k≅k'. In other words we shall see that Gal(A_k/k) does not determine the isomorphism class of an algebraic number field k. Furthermore, the counterexamples which we give will show that even if Gal(A_k/k) and Gal(A_k'/k') are isomorphic, the ideal class groups of k and k' are not necessarily isomorphic.

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著者

• Onabe Midori

  Department Of Mathematics Faculty Of Science Ochanomizu University

関連論文

• On Idele Class Groups of Imaginary Quadratic Fields
• On the Isomorphisms of the Galois Groups of the Maximal Abelian Extensions of Imaginary Quadratic Fields

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