Vtyjkyuk

Let $E/F$ be a field extension and let $A, B$ be $ntimes n$ matrices over $F$. Assume that $A, B$ are similar over $E$, i.e. there exists $Pin mathrmGL_n(E)$ s.t. $B = PAP^-1$. Are $A, B$ also similar over $F$, i.e. can we find $Qin mathrmGL_n(F)$ s.t $B = QAQ^-1$?

This statement is true when $E=mathbbC$ and $F = mathbbQ$. More generally, the statement is true for characteristic 0 fields, and the proof can be found in here. However, the proof doesn't work for the char $p$ case, since there exists nonzero polynomials over a characteristic $p$ field (for example, over $mathbbF_p$) which became zero for any evaluation. (For example, $f(x) = x^p-xin mathbbF_p[x]$ is a nonzero polynomial, but $f(a)=0$ for any $ain mathbbF_p$.) So I want to know whether the theorem is also true for characteristic $p$ case of there's a counterexample. Thanks in advance.

Two square matrices over a field $F$ have the same rational canonical form.
The rational canonical form always has entries in the same field $F$. Therefore
if $A$ and $B$ are similar over the extension field $E$ of $F$ then they
are similar over $F$.