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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Modular field extensions


Authors: Nickolas Heerema and David Tucker
Journal: Proc. Amer. Math. Soc. 53 (1975), 301-306
MSC: Primary 12F15
MathSciNet review: 0401724
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Abstract: Let $ K \supset k$ be fields having characteristic $ p \ne 0$. The following is proved. If $ K$ is algebraic over $ k$ then $ K$ is modular over $ k$ if and only if $ K = S{ \otimes _k}M$ where $ S$ is separably algebraic over $ k$ and $ M$ is purely inseparable, modular. If $ K$ is finitely generated over $ k$ (not necessarily algebraic), then $ K$ is modular over $ k$ if and only if $ K$ where $ M$ is finite, purely inseparable, modular over $ k$, and $ S$ is a finitely generated, separable, extension of $ k$. This leads immediately to the representation $ K = (S{ \otimes _k}M){ \otimes _S}R$ where $ S$ is finite separable over $ k,\;M$ is finite, purely inseparable, modular over $ k$ and $ R$ is a regular finitely generated extension of $ S$, This last representation displays subfields of $ K/k$ related to recently obtained Galois theories. The above results are used to analyze transitivity properties of modularity.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0401724-8
PII: S 0002-9939(1975)0401724-8
Keywords: Modular extension, linear disjointness, separability, tensor product
Article copyright: © Copyright 1975 American Mathematical Society