Generalizations of the primitive element theorem

Christos Nikolopoulos,Panagiotis Nikolopoulos

doi:10.1155/s0161171291000637

Abstract

In this paper we generalize the primitive element theorem to the generation of separable algebras over fields and rings. We prove that any finitely generated separable algebra over an infinite field is generated by two elements and if the algebra is commutative it can be generated by one element. We then derive similar results for finitely generated separable algebras over semilocal rings.

Highlights

In this paper we gemeralize the primitive element theorem to the generation of separable algebras over fields and rings
We prove that any finitely generated separable algebra over an infinite field is generated by two elements and if the algebra is commutative it can be generated by one element
1980 stec’c ict.i c]e: 16,17 i. fRXRZTIC. It is a well known result

Summary

Introduction

We show that a finitely generated separable algebra over an infinite field F is generated by two elements over F. If D is a finitely generated separable division algebra over the there exist units u,d in D such that D F [u, d u d-l]. 2.1: Any separable simple algebra finitely generated over a field is generated over the ground field by two conjugate invertible elements.

Results

Conclusion