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Specialization results in Galois theory

Authors: Pierre Dèbes and François Legrand
Journal: Trans. Amer. Math. Soc. 365 (2013), 5259-5275
MSC (2010): Primary 11R58, 12E30, 12E25, 14G05, 14H30; Secondary 12Fxx, 14Gxx, 14H10
Published electronically: March 25, 2013
MathSciNet review: 3074373
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Abstract: The central topic of this paper is this question: is a given $ k$-étale algebra $ \prod _lE_l/k$ the specialization of a given $ k$-cover $ f:X\rightarrow B$ of the same degree at some unramified point $ t_0\in B(k)$? We reduce it to finding $ k$-rational points on a certain $ k$-variety, which we then study over various fields $ k$ of diophantine interest: finite fields, local fields, number fields, etc. We have three main applications. The first one is the following Hilbert-Grunwald statement. If $ f:X\rightarrow \mathbb{P}^1$ is a degree $ n$ $ \mathbb{Q}$-cover with monodromy group $ S_n$ over $ \overline {\mathbb{Q}}$, and finitely many suitably large primes $ p$ are given with partitions $ \{d_{p,1}, \ldots , d_{p,s_p}\}$ of $ n$, there exist infinitely many specializations of $ f$ at points $ t_0\in \mathbb{Q}$ that are degree $ n$ field extensions with residue degrees $ d_{p,1}, \ldots , d_{p,s_p}$ at each prescribed prime $ p$. The second one provides a description of the separable closure of a PAC field $ k$ of characteristic $ p\not =2$: it is generated by all elements $ y$ such that $ y^m-y\in k$ for some $ m\geq 2$. The third one involves Hurwitz moduli spaces and concerns fields of definition of covers.

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Additional Information

Pierre Dèbes
Affiliation: Laboratoire Paul Painlevé, Mathématiques, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France

François Legrand
Affiliation: Laboratoire Paul Painlevé, Mathématiques, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France

Keywords: Specialization, algebraic covers, twisting lemma, Hilbert's irreducibility theorem, Grunwald's problem, PAC fields, local fields, global fields, Hurwitz spaces
Received by editor(s): September 4, 2011
Received by editor(s) in revised form: January 13, 2012, and January 26, 2012
Published electronically: March 25, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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