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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Maximal separable subfields


Author: Bonnie Page Danner
Journal: Proc. Amer. Math. Soc. 68 (1978), 125-131
MSC: Primary 12F15
DOI: https://doi.org/10.1090/S0002-9939-1978-0460300-4
MathSciNet review: 0460300
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Abstract: If $ L/K$ is a finitely generated separable field extension of characteristic $ p \ne 0$ and M is an intermediate field such that $ L/M$ is inseparable, it is proved there exist subfields S of M maximal with respect to the property that $ L/S$ is separable. These maximal separable subfields, denoted S-subfields for $ L/M$, are characterized in two ways.

(1) Let $ L/S$ be a separable field extension. Then S is a S-subfield for $ L/M$ if and only if $ S({L^p}) \supseteq M$ and S is algebraically closed in M.

(2) If $ L/S$ is separable, S is a S-subfield for $ L/M$ if and only if the inseparability of $ L/M$ is equal to the transcendence degree of $ M/S$.

A S-subfield for $ L/M$ is constructed using a maximal subset of a relative p-basis for $ M/K$ which remains p-independent in L. It is proved that there is a unique S-subfield for $ L/M$ if and only if $ S/K$ is algebraic for some S.


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DOI: https://doi.org/10.1090/S0002-9939-1978-0460300-4
Keywords: Separable and inseparable field extensions, p-bases
Article copyright: © Copyright 1978 American Mathematical Society

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