Alternating group coverings of the affine line for characteristic greater than two
type: Item
instance of: scholarly article
title: Alternating group coverings of the affine line for characteristic greater than two
zbMATH Open document ID: 0801.12003
DOI: 10.1007/BF01445094
published in: Mathematische Annalen
publication date: 1994-11-29T00:00:00Z
full work available at URL: https://eudml.org/doc/165074
review text: Let \(L_ k\) be the affine line over an algebraically closed ground field \(k\) of nonzero characteristic \(p\), let \(q\) be a positive power of \(p\) and let \(n\), \(t\) be two integers with \(n= q+t\), \(t\not\equiv 0\pmod p\). Let the unramified covering of \(L_ k\) be given by \(\overline{F}_{n,q} =0\), where \(\overline{F}_{n,q}= Y^ n- XY^ t +1\) and let \(\overline{G}_{n,q}= \text{Gal} (\overline{F}_{n,q}, k(X))\) be the associated Galois group. By using some arguments about the permutation groups \(A_ n\) (alternating group), \(S_ n\) (symmetric group) and the relations with the Galois theory, the author establishes the main theorems of the paper: Theorem 1. If \(p>2\) and \(q<t\), then \(\overline{G}_{n,q}= A_ n\); Theorem 2. If \(p=q =2\) then \(\overline{G}_{n,q}= S_ n\). If \(2= p<q <t\) then \(\overline{G}_{n,q}= A_ n\) or \(S_ n\). The paper, which also contains other interesting theorems, completes a series of works of the author concerning the subject.
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 20D06
zbMATH DE Number: 537286
zbMATH Keywords: alternating group
zbMATH Keywords: symmetric group
zbMATH Keywords: affine line
zbMATH Keywords: unramified covering
zbMATH Keywords: Galois group
zbMATH Keywords: permutation groups
reviewed by: Toma Albu
MaRDI profile type: Publication
cites work: Local uniformization on algebraic surfaces over ground fields of characteristic \(p\neq 0\)
cites work: Coverings of Algebraic Curves
cites work: Ramification Theoretic Methods in Algebraic Geometry (AM-43)
cites work: Q3491708
cites work: Galois theory on the line in nonzero characteristic
cites work: Q3986644
cites work: Wreath products and enlargements of groups
cites work: Linear Disjointness of Polynomials
cites work: Alternating group coverings of the affine line for characteristic two
cites work: Small degree coverings of the affine line in characteristic two
cites work: Q5512231
author: Shreeram S. Abhyankar
description: scientific article; zbMATH DE number 537286