Some studies on additive commutator groups in division rings (Q5939024)

Let \(D\) be a division ring of characteristic 0, with centre \(F\) and denote by \([D,D]\) the additive subgroup generated by the additive commutators \([a,b]=ab- ba\) \((a,b\in D)\). For any subset \(S\) of \(D\) the \(F\)-algebra generated by \(S\) is denoted by \(F[S]\). The author proves that for any noncommutative division subring \(D_1\), algebraic over \(F\), \(D_1=F[D_1\cap[D,D]]\). In particular, if \(D\) is algebraic over \(F\), then \(D=F[[D,D]]\). Next, if \(K\) is a subfield of \(D\) containing \(F\) and algebraic over \(F\), then \(K=F[G]\) for some subgroup \(G\) of \([D,D]\). If moreover \(D\) is algebraic over \(F\), then the maximal subfields of \(D\) correspond to certain maximal subgroups of \([D,D]\). The author further examines the relations between isomorphic subfields of a division algebra.