Geometric characterization of strongly normal extensions

Kovacic, Jerald

doi:10.1090/S0002-9947-06-03868-2

Geometric characterization of strongly normal extensions
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by Jerald J. Kovacic PDF

Trans. Amer. Math. Soc. 358 (2006), 4135-4157 Request permission

Abstract:

This paper continues previous work in which we developed the Galois theory of strongly normal extensions using differential schemes. In the present paper we derive two main results. First, we show that an extension is strongly normal if and only if a certain differential scheme splits, i.e. is obtained by base extension of a scheme over constants. This gives a geometric characterization to the notion of strongly normal. Second, we show that Picard-Vessiot extensions are characterized by their Galois group being affine. Our proofs are elementary and do not use “group chunks” or cohomology. We end by recalling some important results about strongly normal extensions with the hope of spurring future research.

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