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On the Cyclotomic Unit Group and the Ideal Class Group of a Real Abelian Number Field II

โœ Scribed by Manabu Ozaki

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
1001 KB
Volume
64
Category
Article
ISSN
0022-314X

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โœฆ Synopsis

Let K be a real abelian number field satisfying certain conditions and K n the n th layer of the cyclotomic Z p -extension of K. We study the relation between the p-Sylow subgroup of the ideal class group and that of the unit group module the cyclotomic unit group of K n . We give certain sufficient conditions which assure that the above two groups are isomorphic as Galois modules for sufficiently large n. We shall also show that they have the same p-rank for sufficiently large n.

1997 Academic Press

Theorem. Suppose that K satisfies one of the following conditions:

(i) The conductor of K is the prime p,

Then if the Iwasawa *-invariant of K ร‚K ([6]) vanishes, A n is isomorphic with B n as Galois modules for sufficiently large n.

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Let q be a power of a prime number p and k=F q (T ) the rational function field with a fixed indeterminate T. For an irreducible monic P=P(T ) in R=F q [T], let k(P) + be the maximal real subfield of the P th cyclotomic function field and h + T (P) the class number of k(P) + associated to R. We prov