Noncommutative splitting fields
type: Item
instance of: scholarly article
title: Noncommutative splitting fields
zbMATH Open document ID: 0697.16018
DOI: 10.1016/0021-8693(90)90225-D
author: Jan Treur
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: The author considers extensions of fields \(K\subset N\) (not necessarily commutative) and elements of N which are left zeros of polynomials \(f=\sum x^ ia_ i\) over K. For any subset S of N consider the right ideal of K[x] of polynomials having S among their left zeros; it is principal and if non-zero, its generator is written \(q_ s\). For any polynomial p over K denote by Z(p) the set of its left zeros in N. If \(S\leq Z(p)\) and \(q_ s=p\), then S is called a splitting set for p, and N is a splitting field for p if it is generated over K by a splitting set for p. If for every finite subset F of S and any \(\alpha\in S\setminus F\) there is a polynomial vanishing on F but not for \(\alpha\), then S is called separable. An extension L/K is (inner) Galois if K is the fixed field of some group of (inner) automorphisms of L; a subset of an orbit of the group is called a Galois set. The author notes that in a Galois extension the minimal polynomial of any right algebraic element is split by a Galois set. He is mainly concerned with field coproducts; in a field coproduct, if an element in one factor is right algebraic over another factor, then it is right algebraic over the ground field with the same minimal polynomial. This is applied to show that in a field coproduct, if for a polynomial p over the ground field a left zero is taken in each of several factors (of the field coproduct) then these left zeros form a separable set and if this set has more than \(d=\deg p\) elements, then the coproduct splits p and all its left zeros are conjugate. - Let L/K be of finite right degree, with a right predual \(L_ 1/K_ 1\) in an extension N [cf. the author, J. Algebra 119, 1-22 (1988; Zbl 0661.16015)]. Then any right minimal polynomial of an element of L is split over N by an inner Galois splitting set. Further, if p is the right minimal polynomial of \(\theta\in L\). Then any separable family \(\{\theta_ L\}\) of left zeros of p in N such that \(K(\theta_ i)\cong K(\theta_ j)\…
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 12E15
Mathematics Subject Classification ID: 16S20
zbMATH DE Number: 4142247
zbMATH Keywords: skew field
zbMATH Keywords: extensions of fields
zbMATH Keywords: left zeros of polynomials
zbMATH Keywords: splitting set
zbMATH Keywords: splitting field
zbMATH Keywords: fixed field
zbMATH Keywords: automorphisms
zbMATH Keywords: Galois set
zbMATH Keywords: Galois extension
zbMATH Keywords: minimal polynomial
zbMATH Keywords: field coproducts
zbMATH Keywords: separable set
zbMATH Keywords: inner Galois splitting set
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4142247