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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A Galois theory for inseparable field extensions

Author: Nickolas Heerema
Journal: Trans. Amer. Math. Soc. 154 (1971), 193-200
MSC: Primary 12.40
MathSciNet review: 0269632
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Abstract: A Galois theory is obtained for fields k of characteristic $ p \ne 0$ in which the Galois subfields h are those for which k/h is normal, modular, and for some nonnegative integer r, $ h({k^{{p^{r + 1}}}})/h$ is separable. The related automorphism groups G are subgroups of the group A of automorphisms $ \alpha $ on $ k[\bar X] = k[X]/{X^{{p^{r + 1}}}}k[X]$, X an indeterminate, such that $ \alpha (\bar X) = \bar X$. A subgroup G of A is Galois if and only if G is a semidirect product of subgroups $ {G_k}$ and $ {G_0}$, where $ {G_k}$ is a Galois group of automorphisms on k (classical separable theory) and $ {G_0}$ is a Galois group of rank $ {p^r}$ higher derivations on k (Jacobson-Davis purely inseparable theory). Implications of certain invariance conditions on a Galois subgroup of a Galois group are also investigated.

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Keywords: Higher derivations, normal field extension, modular field extension, purely inseparable field extension, semidirect product, linear disjointness, tensor product
Article copyright: © Copyright 1971 American Mathematical Society