Higher derivation Galois theory of inseparable field extensions. (English) Zbl 0868.12004

Hazewinkel, M. (ed.), Handbook of algebra. Volume 1. Amsterdam: North-Holland. 187-220 (1996).
This survey of Galois theory for inseparable field extensions is divided into four sections. The first section contains results about the structure of inseparable field extensions and draws upon work by S. MacLane [Duke Math. J. 5, 372-393 (1939; Zbl 0021.10102)], J. Dieudonn√© [Summa Brasil Math. 2, 1-20 (1947; Zbl 0039.26704)] and M. E. Sweedler [Ann. Math., II. Ser. 87, 401-410 (1968; Zbl 0168.29203)], among others. The Galois theory for modular, purely inseparable field extensions using derivations and higher derivations is presented in the second section. The approach taken to the subject here and in the subsequent sections is due to N. Heerema and J. K. Deveney, Trans. Am. Math. Soc. 189, 263-274 (1974; Zbl 0282.12102)] and the authors, although reference also is given to the work of M. Gerstenhaber and A. Zaromp [Bull. Am. Math. Soc. 76, 1011-1014 (1970; Zbl 0205.06303)].
In section three, automorphisms of normal, separable, algebraic field extensions are combined with higher derivations of modular, purely inseparable field extensions to give a Galois theory for modular, normal, algebraic field extensions, while section four contains efforts to combine higher derivations of finite rank with higher derivations of infinite rank in one Galois theory.
Sketches of proofs to many theorems stated in this article are included. But a reader, not already knowledgeable in this area of the theory of fields, may not find the sketches helpful; and even the knowledgeable reader may be distracted by a number of misprints. A case in point is the proof of Theorem 3.30, which this reviewer found quite garbled. Nevertheless, the article is comprehensive, includes statements of many significant theorems, and contains an extensive bibliography.
For the entire collection see [Zbl 0859.00011].


12F15 Inseparable field extensions
12-02 Research exposition (monographs, survey articles) pertaining to field theory