Galois theory of algebraic and differential equations. (English) Zbl 0885.12004

Umemura, Hiroshi

Nagoya Math. J. 144, 1-58 (1996).

Let \(L\) be a differential field and \(K\) a differential subfield of \(L\). The author introduces the notion of quasi-automorphic extension, which unifies the notions of strongly normal and Galois extension. He proves that a differential field extension \(L/K\) is quasi-automorphic if and only if (i) the constants field extension \(C_L/C_K\) is finite and (ii) if there exist a model \(X\) with derivation of \(L/K\) and an algebraic group scheme \(G\) over the field of constants \(C_K\) such that \(X\) is a principal homogeneous space of \(G\). Then he proves the main results of Galois theory using the language of the theory of schemes. We note that
(i) The possibility of building an essentially more general theory than the currently existing Galois theories for differential fields has been discussed by I. Kaplansky [see An introduction to differential algebra, Hermann (Paris, 1957; Zbl 0083.03301)].
(ii) Another approach to the building of the theory, which is based on the use of algebraic Lie pseudogroups and automorphic extensions, was developed by J. F. Pommaret [see Differential Galois theory, Mathematics and Its Applications 15, Gordon and Breach (1983; Zbl 0539.12013)].
(iii) When a field of constants of \(K\) is algebraically closed, any quasi-automorphic extension is strongly normal, and for this case new results are not obtained.

Reviewer: N.V.Grigorenko (Kyïv)

MSC:

12H05	Differential algebra
12H20	Abstract differential equations
14L15	Group schemes
12F10	Separable extensions, Galois theory
14M17	Homogeneous spaces and generalizations

Keywords:

strongly normal extension; differential field; quasi-automorphic extension; Galois theory

Citations:

Zbl 0083.03301; Zbl 0539.12013

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