Extension of valuations in skew fields

Abstract

The aim of this thesis is to study the extension of valuations in skew field extensions.

In Chapter I we look at the following problem.

Let K be a field and V a valuation ring of rank 1 in K. Let H be a crossed product division algebra over K. Then we study conditions under which there exists a matrix local ring R in H lying over V and generating H as K-space. We then find that R is a valuation ring in H lying over V iff R is local. Moreover if V is discrete of rank 1, then R is a maximal order in H.

In Chapter II we study directly conditions under which a valuation on the centre of a finite dimensional central division algebra can be extended to the whole algebra. In particular if H = (E/K; sigma, a) is a cyclic division algebra and v is a discrete rank 1 valuation on K, then the extension of v to H depends on v(a). We then carry on the study of the extension problem for the tensor product of algebras. In particular if [special characters omitted] and V a rank 1 valuation ring in K and if there exists a valuation ring W in H lying over V with W ∩ H_i = w_i (i = 1,...,r), we study conditions under which [special characters omitted].

In Chapter III we look at infinite skew field extensions. We study valuations in skew function fields. The application will include among others, free algebras, universal associative envelopes of Lie algebras and generic crossed product. However our main concern in this chapter is the following question raised by P.M. Cohn.

Let K₁, K₂ be two skew fields with a common subfield K and let v₁, v₂ be real valued valuations on K ₁ and K₂ respectively such that v₁|K = v ₂|K = v.

Do v₁, v₂ have a common extension to H = K ₁ O_K K₂ (the field coproduct of K₁ and K₂)?.

We show that in general the answer is no. Nevertheless we find conditions under which v₁,v2 have a common extension to H.