I am wondering what, if anything, is known about the characteristic polynomials of integer symmetric matrices. I believe I read somewhere that not every polynomial with integer coefficients can be a characteristic polynomial for an integer matrix (correct me if I am wrong). What I would like to be true is that basically there aren't really any constraints in some sense.

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Linear and Multilinear Algebra2(1974), 173--178. – Richard Stanley Jan 25 '12 at 23:39