Pre-Galois theory

Harbater, David; Dèbes, Pierre

Document type :

Compte-rendu et recension critique d'ouvrage

DOI :

10.48550/arXiv.1906.06631

Title :

Pre-Galois theory

Author(s) :

Harbater, David [Auteur]
Dèbes, Pierre [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Journal title :

Osaka Journal of Mathematics

Pages :

71--101

Publisher :

Osaka University

Publication date :

2021

ISSN :

0030-6126

English keyword(s) :

Number Theory (math.NT)
FOS: Mathematics
12F (primary)
14H (secondary)

HAL domain(s) :

Mathématiques [math]

English abstract : [en]

We introduce and study a class of field extensions that we call pre-Galois; viz. extensions that become Galois after some linearly disjoint Galois base change L/k. Among them are geometrically Galois extensions of κ(T), ...
Show more >We introduce and study a class of field extensions that we call pre-Galois; viz. extensions that become Galois after some linearly disjoint Galois base change L/k. Among them are geometrically Galois extensions of κ(T), with κ a field: extensions that become Galois and remain of the same degree over κ(T). We develop a pre-Galois theory that includes a Galois correspondence, and investigate the corresponding variants of the inverse Galois problem. We provide answers in situations where the classical analogs are not known. In particular, for every finite simple group G, some power Gn is a geometric Galois group over k, and is a pre-Galois group over k if k is Hilbertian. For every finite group G, the same conclusion holds for G itself (n = 1) if k = Qab and G has a weakly rigid tuple of conjugacy classes; and then G is a regular Galois group over an extension of Qab of degree dividing the order of Out(G). We also show that the inverse problem for pre-Galois extensions over a field k (that every finite group is a pre-Galois group over k) is equivalent to the a priori stronger inverse Galois problem over k, and similarly for the geometric vs. regular variants.Show less >

Language :

Anglais

Popular science :

Non

Collections :

Laboratoire Paul Painlevé - UMR 8524

Source :

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