Algebraic Extensions of Difference Fields

Abstract

An inversive difference field $\mathcal {K}$ is a field K together with a finite number of automorphisms of K. This paper studies inversive extensions of inversive difference fields whose underlying field extensions are separable algebraic. The principal tool in our investigations is a Galois theory, first developed by A. E. Babbitt, Jr. for finite dimensional extensions of ordinary difference fields and extended in this work to partial difference field extensions whose underlying field extensions are infinite dimensional Galois. It is shown that if $\mathcal {L}$ is a finitely generated separable algebraic inversive extension of an inversive partial difference field $\mathcal {K}$ and the automorphisms of $\mathcal {K}$ commute on the underlying field of $\mathcal {K}$ then every inversive subextension of $\mathcal {L}/\mathcal {K}$ is finitely generated. For ordinary difference fields the paper makes a study of the structure of benign extensions, the group of difference automorphisms of a difference field extension, and two types of extensions which play a significant role in the study of difference algebra: monadic extensions (difference field extensions $\mathcal {L}/\mathcal {K}$ having at most one difference isomorphism into any extension of $\mathcal {K}$) and incompatible extensions (extensions $\mathcal {L}/\mathcal {K},\mathcal {M}/\mathcal {K}$ having no difference field compositum).