TY - JOUR
T1 - The Galois group of random elements of linear groups
AU - Lubotzky, Alexander
AU - Rosenzweig, Lior
N1 - Publisher Copyright:
© 2014 by Johns Hopkins University Press.
PY - 2014/10/1
Y1 - 2014/10/1
N2 - 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).
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).
UR - http://www.scopus.com/inward/record.url?scp=85000434645&partnerID=8YFLogxK
U2 - 10.1353/ajm.2014.0038
DO - 10.1353/ajm.2014.0038
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AN - SCOPUS:85000434645
SN - 0002-9327
VL - 136
SP - 1347
EP - 1383
JO - American Journal of Mathematics
JF - American Journal of Mathematics
IS - 5
ER -