On a Theorem of Latimer and Macduffee

Abstract

The matrix solutions of an irreducible algebraic equation with integral coefficients were studied by Latimer and MacDuffee. They considered matrices with rational integers as elements. If A is such a matrix, then all matrices of the “class” S1 AS will again be solutions if S is a matrix of determinant ± 1. On the other hand, in general all solutions cannot be derived in this way from one solution only. It was in fact shown that the number of classes of matrix solutions coincides with the number of different classes of ideals in the ring generated by an algebraic root of the same equation.