Generically algebraic algebras

Abstract

The notions of generic minimum polynomial and generic norm for finitedimensional strictly power-associative algebras were introduced by Professor N. Jacobson in [2], [3], generalizing the notions of principal polynomial and norm for associative algebras. In this paper we extend these concepts to infinite-dimensional algebras; in the process we give a coordinate-free approach to the generic norm. In the first section we review the machinery of the differential calculus in infinite-dimensional spaces, including the definition of rational mappings, the Zariski topology, and differential operators. In the second section we establish the result, which is fundamental in the sequel, that every homogeneous polynomial function can be written uniquely as a product of a finite number of irreducible polynomial functions. We then apply this factorization theory to show that under certain general conditions the factors of an automorphic form (relative to a group of linear transformations) are again automorphic forms. The third and fourth sections are devoted to defining and establishing the basic properties of the generic minimum polynomial and generic norm. A generically algebraic algebra is defined as one which is uniformly algebraic in the sense that each element x satisfies a monic polynomial «i*(A) which varies continuously as a function of x: mx(x)=0 for «ix(A) = 2 ml(x)Xi where the coefficients are polynomial functions. The generic minimum polynomial is the polynomial of least degree having these properties, and the generic norm is plus or minus its constant term. Using the factorization theorem the standard properties of the generic norm carry over easily. The fifth section discusses the discriminant of an algebra. An algebra is called unramified if its discriminant is not identically zero. Intuitively, this means that the algebra modulo or radical is separable. Our definition is motivated by [1, p. 105]; it differs from that given in [2]. Several investigations revealed a close connection between the generic norm and forms admitting composition [4], [5], [6], [7], [8]. A general conjecture of Professor R. D. Schafer was that every norm on a normed algebra was a product of irreducible factors of the generic norm. This was known for finite-dimensional algebras [4] and for infinite-dimensional algebras in certain special cases [5]. In the sixth section of the paper we apply our results to settle the general case.