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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Modular field extensions
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by Nickolas Heerema and David Tucker
Proc. Amer. Math. Soc. 53 (1975), 301-306
DOI: https://doi.org/10.1090/S0002-9939-1975-0401724-8

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.
References
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Bibliographic Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 53 (1975), 301-306
  • MSC: Primary 12F15
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0401724-8
  • MathSciNet review: 0401724