scholarly journals Class groups of real cyclotomic fields

2021 ◽  
Author(s):  
Mohit Mishra ◽  
Rene Schoof ◽  
Lawrence C. Washington
Keyword(s):  
Cyclotomic Fields ◽  
Class Groups

On minus quotients of ideal class groups of cyclotomic fields

2020 ◽  
Vol 16 (09) ◽  
pp. 2013-2026
Author(s):  
Satoshi Fujii
Keyword(s):  
Abelian Group ◽  
Class Group ◽  
Finite Abelian Group ◽  
Cyclotomic Field ◽  
Cyclotomic Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Ideal Class Groups ◽  
The Ideal

Let [Formula: see text] be the minus quotient of the ideal class group of the [Formula: see text]th cyclotomic field. In this paper, first, we show that each finite abelian group appears as a subgroup of [Formula: see text] for some [Formula: see text]. Second, we show that, for all pairs of integers [Formula: see text] and [Formula: see text] with [Formula: see text], the kernel of the lifting map [Formula: see text] is contained in the [Formula: see text]-torsion [Formula: see text] of [Formula: see text]. Such an evaluation of the exponent is an individuality of cyclotomic fields.

scholarly journals An extension theorem for integral representations

1997 ◽  
Vol 63 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Wolfgang Knapp ◽  
Peter Schmid
Keyword(s):  
Finite Group ◽  
Matrix Representation ◽  
Extension Theorem ◽  
Iwasawa Theory ◽  
Integral Representations ◽  
Roots Of Unity ◽  
Cyclotomic Fields ◽  
Qualitative Result ◽  
Class Groups ◽  
A Finite Group

AbstractBy a fundamental theorem of Brauer every irreducible character of a finite group G can be written in the field Q(εm) of mth roots of unity where m is the exponent of G. Is it always possible to find a matrix representation over its ring Z[εm] of integers? In the present paper it is shown that this holds true provided it is valid for the quasisimple groups. The reduction to such groups relies on a useful extension theorem for integral representations. Iwasawa theory on class groups of cyclotomic fields gives evidence that the answer is at least affirmative when the exponent is replaced by the order, and provides for a general qualitative result.

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