On a Galois theory for inseparable field extensions

Mordeson, John N.

doi:10.1090/S0002-9947-1975-0384762-8

On a Galois theory for inseparable field extensions
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by John N. Mordeson

Trans. Amer. Math. Soc. 214 (1975), 337-347

DOI: https://doi.org/10.1090/S0002-9947-1975-0384762-8

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Abstract:

Heerema has developed a Galois theory for fields L of characteristic $p \ne 0$ in which the Galois subfields K are those for which $L/K$ is normal, modular and, for some nonnegative integer $e,K({L^{{p^{e + 1}}}})/K$ is separable. The related automorphism groups G are subgroups of a particular group A of automorphisms on $L[x]/{x^{{p^e} + 1}}L[x]$ where x is an indeterminate over L. For $H \subseteq G$ Galois subgroups of A, we give a necessary and sufficient condition for H to be G-invariant. An extension of a result of the classical Galois theory is also given as is a necessary and sufficient condition for every intermediate field of $L/K$ to be Galois where K is a Galois subfield of L.

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