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.
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机译：设A是无限域k上的（不一定是关联的）代数有限维，而A {x}是不确定x上的多项式函数的代数。 A {x}中的可分离多项式函数定义为在（'）k（CRTIMES）A中具有最大可能有限数量的不同零的多项式函数，其中（'）k是k的代数闭包。图中的k的场扩展L，L（L-HOOK）（'）k最小，其中基础扩展的代数L（CRTIMES）（，k）A包含可分离多项式函数的所有根。是k的Galois扩展。如果A是严格简单的，则通过邻接可分离多项式的所有根获得的代数与L（CRTIMES）（，k）A重合;让M（A）表示有限维代数A的乘法代数。 （A）是半简单的，则M（A）是同构的，是交换场上全矩阵环的乘积。如果M（A）的Jacobson根J为最大理想值，则M（A）/ J在交换场上对全矩阵环是同构的; Gelfand-Kirillov维数（GK维数）的定义和基本性质）扩展到不一定是关联的代数。有限型代数的G-K维数等于其乘积代数左模块的G-K维数。多项式函数的代数在有限维代数上的G-K维数证明是介于零和代数维之间的整数。从所有d维k代数的仿射空间到集合{0,1，...，d}的函数，A {x}的A（--->）G-K维是射影。给出了矢量空间维等于多项式函数代数的G-K维的那些代数的代数表征。
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