Field Theoretic Cogalois Theory via Abstract Cogalois Theory

Albu, Toma

doi:10.1016/j.jpaa.2005.11.008

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Unitarily Graded Field Extensions the Aim Of This Section I1

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“…, and the given grading of L can be identified with the grading of part (1). Furthermore, the only intermediate fields of L|K are the graded fields…”

Section: Unitarily Graded Field Extensions the Aim Of This Section Imentioning

confidence: 99%

Unitarily graded field extensions

2007

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We introduce the universal unitarily graded A-algebra for a commutative ring A and an arbitrary abelian extension U of the group of units of A, and use this concept to give simplified proofs of the main theorems of co-Galois theory in the sense of T. Albu. The main tool is a generalisation of a theorem by M. Kneser which, in our language, is a criterion for the universal algebra to be a field when the base ring A is itself a field. This theorem implies also the theorem of A. Schinzel on linearly independent roots. We discuss examples involving the injective hull of the multiplicative group of a field and we develop criteria for Galois extensions which allow a co-Galois grading, in particular for the cyclic case.Comment: Some minor changes; to appear in Acta Arithmetic

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“…, and the given grading of L can be identified with the grading of part (1). Furthermore, the only intermediate fields of L|K are the graded fields…”

Section: Unitarily Graded Field Extensions the Aim Of This Section Imentioning

confidence: 99%