Noncommutative splitting fields
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 \(K(\theta_ i)\cong K(\theta_ j)\) with \(\theta_ i\leftrightarrow \theta_ j\) can be obtained as \(t_ i^{-1}\theta t_ i\) where the \(t_ i\) form part of a left basis of \(L_ 1/K_ 1\). The paper ends with a number of questions. Sample: when does a polynomial in K have a universal zero (yielding all others by specialization over K)?
scientific article; zbMATH DE number 4142247