Not to mention that when your matrix is 5x5 or more, there aren't even general solutions for roots if for some reason you're still insisting on the matrix->polynomial->eigenvalues route.
Sure there are (in many reasonable senses). Polynomial root extraction just isn't expressible in terms of addition, subtraction, multiplication, division, and nth roots alone. But that's ok; there's nothing magic about that particular set of operations, so as to make it the end-all, be-all.