Algebraic elements in valued *-division rings. (Q1432749)

Let \(D\) be a division ring with an involution *, and \(w\) a *-valuation of \(D\) (i.e. a valuation satisfying the condition \(w(d)=w(d^*)\), for each \(d\in D\)), \(R\) the valuation ring of \((D,^*)\), \(M\) the maximal ideal of \(R\), and \(\widehat D\) the residue field of \((D,^*)\). The valuation \(w\) is said to be invariant, if \(sbs^{-1}\equiv b\pmod M\), provided that \(s,b \in D\), \(s=s^*\), and \(s\) or \(b\) lies in \(R\); \(w\) is called strongly invariant, if the element \(sds^{-1}d^{-1}\) lies in \(M\) whenever \(s,d\in D^*\) and \(s=s^*\). The paper under review shows that if \(D\) is strongly invariant of characteristic different from \(2\), then it is invariant. When \(w\) is invariant and \(\widehat A\) is the image in \(\widehat D^*\) of the subgroup \(A\) of the multiplicative group \(D^*\), generated by all commutators of elements of \(D^*\), at least one of which is algebraic over the centre \(Z\) of \(D\), the author obtains that if \(\mu\) is an element of \(D\) algebraic over \(Z\), and \(p\) is a prime divisor of the degree \([Z(\mu):Z]\), then \(p\) equals \(\text{char}(\widehat D)\) or the order of a suitably chosen element of \(\widehat A\). Assuming finally that \(w\) is invariant and \(D\) is algebraic over \(Z\) (i.e. all elements of \(D\) are algebraic over \(Z\)), he proves that the ramification index of \(w\) relative to the valuation \(w_Z\) of \(Z\) induced by \(w\) is a power of \(2\), the dimension \(f\) of \(\widehat D\) as an algebra over the residue field \(\widehat Z\) of \((Z,w_Z)\) equals \(1\), \(2\) or \(4\), and \(\widehat D\) is noncommutative in case \(f=4\).