Unitarily graded field extensions

Abstract

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.