Characteristic Polynomials

Introduction. Let F be a field and let V be a finite dimensional vector space over F which is also a module over the ring F[a]. Here a may lie in any extension ring of F. We do not assume, as yet, that V is a faithful module, so that a need not be a linear transformation on V. It is known that by means of a decomposition of V into cyclic F[a]-modules we may obtain a definition of the characteristic polynomial of a on V which does not involve determinants. In this note we shall give another non-determinantal definition of the characteristic polynomial. Instead of considering a single module V, we shall accordingly study the set of all finite dimensional F[a]-modules and mappings of this set into monic polynomials with coefficients in F. Admittedly our procedure does not yield the theory of the elementary divisors of a, but it has certain advantages. First , all questions of uniqueness are settled immediately by the Jordan-Holder theorem. Secondly, it is possible to derive some classical results, usually proved using determinants, without excessive labour. To illustrate the use of our method we shall complete and generalise some results due to Goldhaber (2) and Osborne (5).