Characteristic polynomial of a symmetric integer matrix

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.

flag	asked Jan 25 2012 at 21:47 Elena 265●1●5

For starters, it cannot have any complex roots. – Yoav Kallus Jan 25 2012 at 22:00

Yoav... complex? That's quite a restriction... :P – darij grinberg Jan 25 2012 at 22:36

He means that the eigenvalues of a real symmetric matrix are real. – Tom Goodwillie Jan 25 2012 at 22:43

Think of a simple case: $2\times 2$ symmetric matrices with trace zero. The question becomes, which integers may be expressed as the sum of two squares? The answer is well-known, and yes there are constraints! – Tom Goodwillie Jan 25 2012 at 22:56

Some information is in the paper E. Bender and N. P. Herzberg, Linear and Multilinear Algebra 2 (1974), 173--178. – Richard Stanley Jan 25 2012 at 23:39

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