Modular field extensions

Heerema, Nickolas; Tucker, David

doi:10.1090/S0002-9939-1975-0401724-8

Modular field extensions
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by Nickolas Heerema and David Tucker PDF

Proc. Amer. Math. Soc. 53 (1975), 301-306 Request permission

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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