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Ideal Class Groups of Iwasawa‐Theoretical Abelian Extensions Over the Rational Field
Author(s) -
Horie Kuniaki
Publication year - 2002
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610702003502
Subject(s) - ideal class group , mathematics , class field theory , algebraic number field , ideal (ethics) , prime ideal , abelian group , algebraic number , field (mathematics) , iwasawa theory , algebraic number theory , prime (order theory) , ring of integers , prime number , combinatorics , discrete mathematics , pure mathematics , mathematical analysis , philosophy , epistemology
Throughout this paper, we shall suppose that all algebraic number fields, namely, all algebraic extensions over the rational field Q, are contained in the complex field C. Let P be the set of all prime numbers. For any algebraic number field F , let C F denote the ideal class group of F and, writing F + for the maximal real subfield of F , letC F −denote the kernel of the norm map from C F to the ideal class group of F + ; for each l ∈ P , let C F ( l ) denote the l ‐class group of F , that is, the l ‐primary component of C F , and letC F − ( l )denote the l ‐primary component ofC F − . Furthermore, for each l ∈ P , we denote by Z l the ring of l ‐adic integers.