Minimal polynomials of integral symmetric matrices

Abstract

We investigate what the possible minimal polynomials are for integral symmetric matrices. We show that the obvious necessary conditions are sufficient for polynomials of degree at most 4. We show that necessary conditions are sufficient for minimal polynomials of self-adjoint operators on positive definite unimodular lattices. We also give a relatively elementary proof of the result of Estes that any total real algebraic integer is the eigenvalue of an integral symmetric matrix. This question was asked by Alan Hoffman, who also showed that this result implies that any totally real algebraic integer is the eigenvalue of the adjacency matrix of some graph.