On additive commutator groups in division rings
Let \(D\) be a division ring with centre \(F\). The authors study algebraicity of \(D/F\) in terms of \([D,D]\), the additive group generated by the additive commutators. If \(D\) is finite-dimensional over \(F\), the reduced trace shows that \(D/[D,D]\) is an \(F\)-space of dimension at most one. In characteristic zero this is also shown to hold when \(D\) is algebraic over \(F\). Next, the authors prove the following generalization of Jacobson's theorem [cf. \textit{N. Jacobson}, Ann. Math., II. Ser. 46, 695-707 (1945; Zbl 0060.07501)]: If for any \(x,y\in D\), \(xy-yx\) is algebraic of degree at most \(n\) (where \(n\) is fixed), then \([D:F]\leq(n+1)^4/4\). They conjecture that here the last bound may be replaced by \(n^2\). The rest of the paper brings a number of special results on separable subfields and primitive elements. Sample: If \(K\) is a finite separable extension of \(F\) in \(D\) without a primitive element in \([D,D]\), then \(K\) has exactly one maximal subfield lying in \([D,D]\).
scientific article; zbMATH DE number 1140101
scientific article