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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Galois theory for fields $ K/k$ finitely generated


Authors: Nickolas Heerema and James Deveney
Journal: Trans. Amer. Math. Soc. 189 (1974), 263-274
MSC: Primary 12F15; Secondary 12F10
MathSciNet review: 0330124
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Abstract: Let K be a field of characteristic $ p \ne 0$. A subgroup G of the group $ {H^t}(K)$ of rank t higher derivations $ (t \leq \infty )$ is Galois if G is the group of all d in $ {H^t}(K)$ having a given subfield h in its field of constants where K is finitely generated over h. We prove: G is Galois if and only if it is the closed group (in the higher derivation topology) generated over K by a finite, abelian, independent normal iterative set F of higher derivations or equivalently, if and only if it is a closed group generated by a normal subset possessing a dual basis. If $ t < \infty $ the higher derivation topology is discrete. M. Sweedler has shown that, in this case, h is a Galois subfield if and only if $ K/h$ is finite modular and purely inseparable. Also, the characterization of Galois groups for $ t < \infty $ is closely related to the Galois theory announced by Gerstenhaber and and Zaromp. In the case $ t = \infty $, a subfield h is Galois if and only if $ K/h$ is regular. Among the applications made are the following: (1) $ { \cap _n}h({K^{{p^n}}})$ is the separable algebraic closure of h in K, and (2) if $ K/h$ is algebraically closed, $ K/h$ is regular if and only if $ K/h({K^{{p^n}}})$ is modular for $ n > 0$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0330124-8
PII: S 0002-9947(1974)0330124-8
Keywords: Higher derivation, iterative higher derivation, dual basis, Galois group of higher derivations, independent abelian sets of higher derivations
Article copyright: © Copyright 1974 American Mathematical Society