Eigenvalues of operator given polynomial I have a homework question that is really bothering me because I cannot manage to use any of the theorems we have studied so far, it seems. Could someone please give me a pointer please (no, Cayley-Hamilton is not the answer) ? I can't quite tie it to the characteristic polynomial, unless I am being stupid; this is over a generic field, not necessarily reals or complex.
Let $p(t) = t^2 + t + 1$, and let it be given that $p(T) = 0$ for an operator $T$. List all the eigenvalues of $T$.

As far as p(T)=0, this means that characteristic f(T) pol. divides it: f|p (I mean that p=f*g, g:pol.)  Thus your eigenvalue is one of the roots of p or both of them, as far as g=id.
