On the Isomorphisms of the Galois Groups of the Maximal Abelian Extensions of Imaginary Quadratic Fields  [in Japanese] On the Isomorphisms of the Galois Groups of the Maximal Abelian Extensions of Imaginary Quadratic Fields  [in Japanese]

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Abstract

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.

Journal

  • Natural science report of the Ochanomizu University

    Natural science report of the Ochanomizu University 27(2), p155-161, 1976-12

    Ochanomizu University

Keywords

Codes

  • NII Article ID (NAID)
    110006559055
  • NII NACSIS-CAT ID (NCID)
    AN00033958
  • Text Lang
    JPN
  • Article Type
    departmental bulletin paper
  • Journal Type
    大学紀要
  • ISSN
    00298190
  • NDL Article ID
    1739305
  • NDL Source Classification
    MA2(整数論)
  • NDL Source Classification
    ZM2(科学技術--科学技術一般--大学・研究所・学会紀要)
  • NDL Call No.
    Z14-182
  • Data Source
    NDL  NII-ELS  IR 
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