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IDEAL CLASS GROUPS OF IWASAWA-THEORETICAL ABELIAN EXTENSIONS OVER THE RATIONAL FIELD

Published online by Cambridge University Press:  24 March 2003

KUNIAKI HORIE
Affiliation:
Department of Mathematics, Tokai University, 1117 Kitakaname, Hiratsuka, 259-1292, Japan
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Abstract

Throughout this paper, we shall suppose that all algebraic number fields, namely, all algebraic extensions over the rational field ${\bb Q}$ , are contained in the complex field ${\bb C}$ . 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 $F$ , let $C^-_F$ denote the kernel of the norm map from $C_F$ to the ideal class group of $F^+$ ; for each $l \in P$ , let $C_F(l)$ denote the $l$ -class group of $F$ , that is, the $l$ -primary component of $C_F$ , and let $C^-_F(l)$ denote the $l$ -primary component of $C^-_F$ . Furthermore, for each $l \in P$ , we denote by ${\bb Z}_l$ the ring of $l$ -adic integers.

Type
Research Article
Copyright
The London Mathematical Society, 2002

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