Galois stability, integrality and realization fields for representations of finite Abelian groups

D. A. Malinin

doi:10.1023/A:1023254607196

Galois stability, integrality and realization fields for representations of finite Abelian groups

D. A. Malinin

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

For a given field F of characteristic 0 we consider a normal extension E/F of finite degree d and finite Abelian subgroups G ⊂ GL_n(E) of a given exponent t. We assume that G is stable under the natural action of the Galois group of E/F and consider the fields E = F(G) that are obtained via adjoining all matrix coefficients of all matrices g ε G to F. It is proved that under some reasonable restrictions for n, any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers, there are only finitely many fields E = F(G) for prescribed integers n and t or prescribed n and d.

Original language	English
Pages (from-to)	215-237
Number of pages	23
Journal	Algebras and Representation Theory
Volume	6
Issue number	2
DOIs	https://doi.org/10.1023/A:1023254607196
Publication status	Published - May 2003

Keywords

Algebraic integers
Galois algebras
Galois group
Integral representations

ASJC Scopus subject areas

Mathematics(all)

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10.1023/A:1023254607196

Cite this

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title = "Galois stability, integrality and realization fields for representations of finite Abelian groups",

abstract = "For a given field F of characteristic 0 we consider a normal extension E/F of finite degree d and finite Abelian subgroups G ⊂ GLn(E) of a given exponent t. We assume that G is stable under the natural action of the Galois group of E/F and consider the fields E = F(G) that are obtained via adjoining all matrix coefficients of all matrices g ε G to F. It is proved that under some reasonable restrictions for n, any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers, there are only finitely many fields E = F(G) for prescribed integers n and t or prescribed n and d.",

keywords = "Algebraic integers, Galois algebras, Galois group, Integral representations",

author = "Malinin, {D. A.}",

note = "Funding Information: The author had the opportunity to work on this subject during his stay at the Max-Planck-Institut f{\"u}r Mathematik and would like to express his gratitude for hospitality. The author is also grateful to DAAD for partial financial support. The author is grateful to the referee for his useful suggestions.",

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T1 - Galois stability, integrality and realization fields for representations of finite Abelian groups

AU - Malinin, D. A.

N1 - Funding Information: The author had the opportunity to work on this subject during his stay at the Max-Planck-Institut für Mathematik and would like to express his gratitude for hospitality. The author is also grateful to DAAD for partial financial support. The author is grateful to the referee for his useful suggestions.

PY - 2003/5

Y1 - 2003/5

N2 - For a given field F of characteristic 0 we consider a normal extension E/F of finite degree d and finite Abelian subgroups G ⊂ GLn(E) of a given exponent t. We assume that G is stable under the natural action of the Galois group of E/F and consider the fields E = F(G) that are obtained via adjoining all matrix coefficients of all matrices g ε G to F. It is proved that under some reasonable restrictions for n, any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers, there are only finitely many fields E = F(G) for prescribed integers n and t or prescribed n and d.

AB - For a given field F of characteristic 0 we consider a normal extension E/F of finite degree d and finite Abelian subgroups G ⊂ GLn(E) of a given exponent t. We assume that G is stable under the natural action of the Galois group of E/F and consider the fields E = F(G) that are obtained via adjoining all matrix coefficients of all matrices g ε G to F. It is proved that under some reasonable restrictions for n, any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers, there are only finitely many fields E = F(G) for prescribed integers n and t or prescribed n and d.

KW - Algebraic integers

KW - Galois algebras

KW - Galois group

KW - Integral representations

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DO - 10.1023/A:1023254607196

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VL - 6

SP - 215

EP - 237

JO - Algebras and Representation Theory

JF - Algebras and Representation Theory

IS - 2

ER -

Galois stability, integrality and realization fields for representations of finite Abelian groups

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Keywords

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