All matrices over a field having a given characteristic polynomial are similar if and only if the polynomial is squarefree

Let F be a field and let p(x) be a polynomial over F whose roots all lie in K. Prove that all matrices over F having characteristic polynomial p(x) are similar if and only if p(x) is squarefree.


Suppose p(x) has a repeated factor. Then there are at least two possible lists of elementary divisors whose product is p(x); in particular there exist two matrices with characteristic polynomial p(x) which are not similar. So if all matrices with characteristic polynomial p(x) are similar, then p(x) has no repeated factors.

Conversely, suppose p(x) has no repeated factors. then there is only one possible list of elementary divisors, and so any two matrices having characteristic polynomial p(x) are similar.

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