Galois theory for fields finitely generated
Authors:
Nickolas Heerema and James Deveney
Journal:
Trans. Amer. Math. Soc. 189 (1974), 263274
MSC:
Primary 12F15; Secondary 12F10
MathSciNet review:
0330124
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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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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403301248
PII:
S 00029947(1974)03301248
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
