Basic noncommutative Galois theory
type: Item
instance of: scholarly article
title: Basic noncommutative Galois theory
zbMATH Open document ID: 0826.16033
published in: Advances in Mathematics
publication date: 1995-11-13T00:00:00Z
review text: Let \(K \subseteq L\) be a pair of skewfields such that for every \(\ell \in L - K\) there exists some automorphism \(g:L\to L\) of \(L\) with \(g(k) = k\) for all \(k \in K\), but \(g (\ell) \neq \ell\). In this case \(L\) is called a Galois extension of the skewfield \(K\), while the group \(\text{Aut}_K(L)\), consisting of all automorphisms of \(L\) which keep every element in \(K\) fixed, will be called the Galois group of that Galois extension. A Galois extension \(L\) of a skewfield \(K\) is said to be finite if the dimension \(\dim_K (L_K)\) of \(L\), considered as a right \(K\)-vector space, is finite. Using the Wedderburn-Artin theory of simple artinian rings, the author gives a proof for the following well-known theorem: For every finite Galois extension \(L\) of a skewfield \(K\) with Galois group \(G\) there exists a canonical one-to-one correspondence between (a) subskewfields \(F\) of \(L\) containing \(K\), on the one hand, and (b) those subgroups \(U\) of \(G\), on the other hand, which satisfy the following additivity condition: (Add) If \(\ell_1\), \(\ell_2\), and \(\ell_1 + \ell_2\) are nonzero elements in \(L\) and if both conjugation by \(\ell_1\) and conjugation by \(\ell_2\) are contained in \(U\), then conjugation by \(\ell_1 + \ell_2\) is also contained in \(U\).
reviewed by: Iuliu Crivei
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16K40
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 12F05
Mathematics Subject Classification ID: 06A15
Mathematics Subject Classification ID: 20B27
zbMATH DE Number: 745279
zbMATH Keywords: skewfields
zbMATH Keywords: Galois extensions
zbMATH Keywords: automorphisms
zbMATH Keywords: Galois groups
zbMATH Keywords: Wedderburn- Artin theory
zbMATH Keywords: simple Artinian rings
zbMATH Keywords: finite Galois extensions
author: Andreas W. M. Dress
MaRDI profile type: Publication
full work available at URL: aima.1995.1006
OpenAlex ID: W2043394194
DOI: 10.1006/AIMA.1995.1006
description: scientific article; zbMATH DE number 745279