The Galois group of random elements of linear groups

Alexander Lubotzky; Lior Rosenzweig

doi:10.1353/ajm.2014.0038

The Galois group of random elements of linear groups

Alexander Lubotzky, Lior Rosenzweig

Einstein Institute of Mathematics

The Hebrew University of Jerusalem

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

Abstract

Let 𝔽 be a finitely generated field of characteristic zero and Γ ≤ GL_n(𝔽) a finitely generated subgroup. For γ ∈ Γ, let Gal(𝔽(γ)/𝔽) be the Galois group of the splitting field of the characteristic polynomial of γ over 𝔽. We show that the structure of Gal(𝔽(γ)/𝔽) has a typical behavior depending on 𝔽, and on the geometry of the Zariski closure of Γ (but not on Γ).

Original language	English
Pages (from-to)	1347-1383
Number of pages	37
Journal	American Journal of Mathematics
Volume	136
Issue number	5
DOIs	https://doi.org/10.1353/ajm.2014.0038
State	Published - 1 Oct 2014

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1353/ajm.2014.0038

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AB - Let 𝔽 be a finitely generated field of characteristic zero and Γ ≤ GLn(𝔽) a finitely generated subgroup. For γ ∈ Γ, let Gal(𝔽(γ)/𝔽) be the Galois group of the splitting field of the characteristic polynomial of γ over 𝔽. We show that the structure of Gal(𝔽(γ)/𝔽) has a typical behavior depending on 𝔽, and on the geometry of the Zariski closure of Γ (but not on Γ).

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M3 - مقالة

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SP - 1347

EP - 1383

JO - American Journal of Mathematics

JF - American Journal of Mathematics

IS - 5

ER -

The Galois group of random elements of linear groups

Abstract

All Science Journal Classification (ASJC) codes

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