TITLE: Is simultaneous similarity of matrices independent from the base field?
QUESTION [7 upvotes]: Suppose that $F$ is a subfield of a field $E$ and, for
$n\times n$ matrices $A_1,\dots,A_m, B_1,\dots,B_m$
over $F$, there exists a matrix $T\in{\rm GL}_n(E)$
such that $T^{-1}A_iT=B_i$ for all $i$.

Does this imply that such a matrix $T$ can be chosen from ${\rm GL}_n(F)$?

It is easy to see that the answer is

yes if $m=1$;
and yes if the field $F$ is infinite.

REPLY [5 votes]: This question is answered in comments:

"As everyone is saying, this follows from Noether-Deuring.
See mathoverflow.net/questions/28469/hilbert-90-for-algebras for a quick proof.
I also asked this question a while back math.stackexchange.com/questions/305696."

        –
David E Speyer
Thanks to all!