Algebraic elements in division rings (Q1924610)

An element \(a\) of a division ring \(D\) is said to be algebraic, if \(a\) is algebraic over the centre \(Z\) of \(D\). In this paper a valuation theoretically criterion is proved to describe the algebraic elements of a division ring. For instance, a sufficient condition is given to ensure that each element of \(D\) which is algebraic must be in \(Z\), i.e. \(Z\) is algebraically closed in \(D\). This result can be applied to ordered division rings as well as the universal envelope of a Lie algebra.