I know a proof of this for $\mathbb C/\mathbb R$, which I assume should hold for extensions of the form $L=K(\alpha)$, $\alpha$ algebraic, $K$ infinite. What is a minimal set of conditions on $L/K$ for this to hold?
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3It always holds (e.g. due to the uniqueness of rational canonical form); see Similar matrices and field extensions. – user580918 Jun 24 '22 at 19:14
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1I think this holds for any field extension. By the Jordan normal form, (I think) $A$ is similar to $B$ over a field $K$ iff for any $P \in K[x]$, $P(A)$ and $P(B)$ have the same rank, which concludes. – Aphelli Jun 24 '22 at 19:15