Latimer-MacDuffee theorem

The Latimer-MacDuffee theorem is a result in abstract algebra, published in 1933 by Claiborne Latimer and Cyrus Colton MacDuffee in a joint paper in the Annals of Mathematics. It concerns a monic, irreducible polynomial f of degree n with integer coefficients. The theorem sets up a one-to-one correspondence between two kinds of object: the similarity classes of n by n integer matrices whose characteristic polynomial is f, where similarity is taken over the integers, and the ideal classes of the order formed by the integer polynomials modulo f, where two ideals count as equivalent when they differ only by a nonzero rational scalar multiple. That order need not be the full ring of integers of the number field, so its ideals need not be invertible. Because such an order has only finitely many ideal classes, the theorem implies that only finitely many integer matrix conjugacy classes share a given characteristic polynomial. Later work by Olga Taussky-Todd made significant contributions to the theory built around it.