Nonassociative cyclic extensions of fields and central simple algebras

TL;DR

This paper introduces nonassociative cyclic extensions of fields and central simple algebras, expanding classical cyclic extension theory into the nonassociative realm with new constructions and generalizations.

Contribution

It defines nonassociative cyclic extensions and constructs them using generalized cyclic division algebras, extending Amitsur's classical results to nonassociative structures.

Findings

Nonassociative cyclic extensions of degree m and q are constructed.

Suitable roots of unity enable the construction of these extensions.

Classical associative cyclic extension results are recovered as special cases.

Abstract

We define nonassociative cyclic extensions of degree m of both fields and central simple algebras over fields. If a suitable field contains a primitive mth (resp., qth) root of unity, we show that suitable nonassociative generalized cyclic division algebras yield nonassociative cyclic extensions of degree m (resp., qs). Some of Amitsur's classical results on non-commutative associative cyclic extensions of both fields and central simple algebras are obtained as special cases.