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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Algebraic extensions of difference fields


Author: Peter Evanovich
Journal: Trans. Amer. Math. Soc. 179 (1973), 1-22
MSC: Primary 12H10
MathSciNet review: 0314809
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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).


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DOI: https://doi.org/10.1090/S0002-9947-1973-0314809-4
Keywords: Partial and ordinary difference fields, algebraic extensions, Galois groups of difference field extensions, finitely generated extensions, limit groups, benign extensions, universal compatibility, monadicity, difference Galois groups, core of the Galois group of a difference field extension
Article copyright: © Copyright 1973 American Mathematical Society