Generation of separable and central simple algebras

Abstract

It is a well-known result of Albert’s [1] that every finite dimensional separable associative algebra A over a field Φ can be generated by two elements ξ, η. In fact, Albert has proved that A has a basis consisting of powers ξiηj. Subsequently Kasch [6] proved that A can be generated by two elements which are conjugate (η = α−1 ξα). Albert’s proof is based on the verification of the property for the complete matrix algebra together with a field extension argument. Kasch’s proof depends on a similar result for division algebras, which makes use of the Galois theory of division rings [5]. It has been noted also by Amitsur [2] that Albert’s (and even Kasch’s) result can be obtained by using the theory of algebras satisfying polynomial idenlities. In the present Note we shall give still another treatment of these questions. Our method is based on one of the central results of the theory of Frobenius algebras and yields a sharpening of the earlier results. In particular, we shall obtain some information on the manner in which a central division algebra is generated by any of its maximal subfields. This extends an earlier result of the author’s [3] concerning maximal subfields which are separable.