Free fields in complete skew fields and their valuations (Q1810544)

Let \(D\) be a skew field, \(K\) a subfield and \(X\) a nonempty set. The free \(D\)-ring \(D_K\langle X\rangle\) over \(K\) on \(X\) is defined to be the ring generated by the union \(D\cup X\), and subject to the defining relations \(kx=xk\), for all \(k\in K\), \(x\in X\). It is known that \(D_K\langle X\rangle\) is a free ideal ring and has a universal fraction field, denoted by \(D_K(X)\), and called a free field. Henceforth, we assume that \(D\), the centre \(C\) of \(D\), and \(X\) are countable sets, and \(K\) is a bicentral subfield of \(D\), i.e. \(K=C_D(C_D(K'))\), where \(C_D(S)\) is the centralizer in \(D\) of an arbitrary nonempty subset \(S\) of \(D\). The paper under review shows that if \(D\) has a discrete valuation \(v\) acting nontrivially on \(K\), and the left \(K\)-vector space \(KcK'\) is infinite-dimensional, for each \(c\in D\setminus\{0\}\), then \(D_K(X)\) embeds in the completion of \(D\) with respect to the topology induced by \(v\); in particular, this implies that \(D_K(X)\) embeds in the Laurent formal power series skew field \(D( (Z))\) in an indeterminate \(Z\) over \(D\). The result obtained provides a new method of constructing valuations on free fields. To begin with, the author proves that every valuation \(w\) of \(D\) is extendable to a valuation of \(D_K(X)\). Furthermore, he shows that if \(K=C\), then the value group of the prolongation of \(w\) on \(D_K(X)\) is Abelian, provided that the value group of \(w\) has the same property. Also, it is proved that if \(K\) is commutative and \(G\) is a finitely or countably generated totally ordered nonabelian group, then \(K_K(X)\) has a valuation acting trivially on \(K\), whose value group is the direct sum \(\mathbb{Z}\times G\), where \(\mathbb{Z}\) is the additive group of integers, and \(\mathbb{Z}\times G\) is considered with the lexicographic ordering.