Galois theory for fields finitely generated

Authors:
Nickolas Heerema and James Deveney

Journal:
Trans. Amer. Math. Soc. **189** (1974), 263-274

MSC:
Primary 12F15; Secondary 12F10

DOI:
https://doi.org/10.1090/S0002-9947-1974-0330124-8

MathSciNet review:
0330124

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Abstract | References | Similar Articles | Additional Information

Abstract: Let *K* be a field of characteristic . A subgroup *G* of the group of rank *t* higher derivations is Galois if *G* is the group of all *d* in 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 the higher derivation topology is discrete. M. Sweedler has shown that, in this case, *h* is a Galois subfield if and only if is finite modular and purely inseparable. Also, the characterization of Galois groups for is closely related to the Galois theory announced by Gerstenhaber and and Zaromp. In the case , a subfield *h* is Galois if and only if is regular. Among the applications made are the following: (1) is the separable algebraic closure of *h* in *K*, and (2) if is algebraically closed, is regular if and only if is modular for .

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DOI:
https://doi.org/10.1090/S0002-9947-1974-0330124-8

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