Find the minimal polynomial given

where

I found the associated matrix, which is

Now I need to compute the characteristic polynomial, and this is

Now suppose I take

then I make

if this is zero, then that polynomial is the minimal one right? If is not zero, then I consider the characteristic polynomial, and that's actually the minimal polynomial because of simple application of Cayley - Hamilton Theorem.

== Yup, but in fact it's easier: since the minimal and the char. polynomials have the same irreducible factors, you know the min. pol. = the char. pol. in this case!

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Another question I have, suppose we're computing the characteristic polynomial, when is it equal to the minimal one?