| Let A be a (not necessarily associative) algebra finite dimensional over an infinite field k, and A{x} its algebra of polynomial functions in the indeterminate x. Separable polynomial functions in A{x} are defined as polynomial functions which have the maximum possible finite number of distinct zeros in (')k (CRTIMES) A where (')k is an algebraic closure of k. The field extension L of k, L (L-HOOK) (')k, minimal among those for which the base extended algebra L (CRTIMES)(,k) A contains all of the roots of a separable polynomial function, is shown to be a Galois extension of k. If A is strictly simple, the algebra obtained by adjoining all of the roots of a separable polynomial coincides with L (CRTIMES)(,k) A.;Let M(A) denote the multiplication algebra of a finite dimensional algebra A. If M(A) is semi-simple, then M(A) is isomorphic to a product of full matrix rings over commutative fields. If the Jacobson radical, J, of M(A) is a maximal ideal, then M(A)/J is isomorphic to a full matrix ring over a commutative field.;The definition and basic properties of Gelfand-Kirillov dimension (G-K dimension) are extended to algebras that are not necessarily associative. The G-K dimension of a finite type algebra is shown to be equal to its G-K dimension as a left module over its multiplication algebra. The G-K dimension of the algebra of polynomial functions in one indeterminate over a finite dimensional algebra is proved to be an integer between zero and the dimension of the algebra. As a function from the affine space of all d-dimensional k-algebras to the set {0,1,...,d}, A (--->) G-K dimension of A{x} is surjective. An algebraic characterization of those algebras whose vector space dimension is equal to the G-K dimension of its algebra of polynomial functions is given. |
