Central extensions of ordered skew fields
This paper deals with central (algebraic) extensions of ordered skew fields and is mainly concerned with the following question: If \(D\) is an ordered skew field with centre \(K\) and if \(F\) is an algebraic, ordered (commutative) field extension of \(K\), is \(D':= F\otimes_ K D\) a skew field having an ordering that extends the ordering of \(F\) resp. \(D\)? It is proved that each ordered skew field can be enlarged to an ordered skew field with real closed centre and this result provides a complete answer of this question in the affirmative. The major tool that is used is valuation theory. The relation between orderings and valuations of commutative fields goes back to Baer and Krull and was generalized to ordered skew fields by P. Conrad and A. Tschimmel. The aim of this paper is to show that for each ordered skew field \(D\) with centre \(K\) there exists a commutative algebraic extension field \(F\) of \(K\) such that the following hold: i) \(D'= F\otimes_ K D\) is a skew field. ii) \(D'\) has an ordering that extends the ordering of \(D\). iii) The natural valuation of \(F\) is henselian with real closed residue (skew) field and divisible value group. By Prestel's theorem \(F\) is real closed, i.e. \(F\) is the real closure of \(K\), and one obtains the following theorem: If \(D\) is an ordered skew field with centre \(K\) and if \(F\) is the real closure of \(K\) then \(F\otimes_ K D\) is a skew field having an ordering that extends the ordering of \(D\). Since \(F(i)\) is algebraically closed, this result provides the following corollary: If \(D\) is an ordered skew field with centre \(K\) and if \(A\) is the algebraic closure of \(K\) then \(A\otimes_ K D\) is a skew field. Finally, this corollary can also be stated as: If \(D\) is an ordered skew field with centre \(K\) and if \(f(x)\in K[x]\) is irreducible over \(K\) then \(f(x)\) is irreducible in \(D[x]\).
scientific article; zbMATH DE number 561306
scientific article