Linear associative algebras in a small number of units, with coordinates ranging over the field C of ordinary complex numbers, have been completely tabulated; that is, their multiplication tables have been reduced to very simple forms. But if we have before us a linear associative algebra, the chances are that its multiplication table would not be in any of the tabulated forms, nor even in such a form that we could readily ascertain to which standard form it was equivalent. Accordingly the question naturally arises: "Can we find invariantive criteria which will tell us when two algebras are equivalent? or, as we say, which will completely characterize the algebras?"
In a previous paper [Annals of Mathematics, Vol. XVI, p.1] we considered the problem of finding invariants which would complete characterize linear associative algebras in two or three units with a modulus, over the field C. The terms "invariants" and "characterize," be it understood, are used here in the sense defined by Professor Dickson. For these special cases, the algebras can be completely characterized by invariants obtained by the application of the following theorem: "In a general n-ary linear algebra over any field F, both characteristic determinants are absolute covariants of the algebra; their coefficients are absolute covariants; and the invariants and covariants of the characteristic determinants and of the coefficients of the powers of ω in these determinants are respectively invariants and covariants of the linear algebra."
If now we proceed to the characterization of the quaternary associative algebras with a modulus, we find that the invariants given us by this theorem only partially characterize these algebras. In other words, the general problem can not be solved by use of this theorem alone.
But fortunately Cartan and Wedderburn have some general theorems which bear on the problem. These tell us that we can characterize the general algebras if we can characterize three special kinds of algebras; namely, simple matric algebras, division algebras, and nilpotent algebras. Now in a given number of units there is only one simple matric algebra. Over the field C, there is no division algebra other than the algebra of ordinary complex numbers; and in general, the Galois Fields are the only algebras of a finite number of elements such that every number except zero has an inverse, whereas, over the field C, there is an infinite number of classes of nilpotent algebras.
In this paper, we shall see how nilpotent algebras can be characterized by the aid of certain homogeneous polynomials whose coefficients are constants of multiplication. For algebras in a small number of units, these polynomials are sufficient if we assume the commutative and associative laws; but to characterize the general nilpotent algebra, we need further invariants.