TITLE: Let A be a square matrix over $\mathbb{C} $ and let B be its Jordan canonical form then are A and B similar over a proper subfield of $\mathbb{C}$?
QUESTION [1 upvotes]: Let A be an $ n\times n $ matrix over $\mathbb {C} $ and let B be its Jordan canonical form , then A and B are similar over $\mathbb {C} $. Now suppose K be a proper subfield of $\mathbb {C} $ , in general A and B are not  similar over K .
But If the entries of A also belong to K and the characteristic polynomial of A splits over K , then can I say that  A and B are similar over K?
Actually, I am supposed  to  find a condition so that A and B would be  similar over any proper subfield of  $\mathbb {C} $.
Any help would  be appreciated. Thanks in advance.

REPLY [2 votes]: If $K$ is a subfield of $\mathbb{C}$ which contains the entries of $A$ and $B$, then $A$ and $B$ are guranteed to be similar on the smaller field $K$. You do not need the hypothesis of the characteristic polynomial splitting over $K$ (if you assume this, you can give a proof by using the fact that $A$ will have a Jordan canonical form $J_{K}$ over $K$ and then showing that $J_K$ and $B$ only differ by a permutation of the blocks) Otherwise, in the general case, you can make use of the Rational canonical form; refer to the answer in this post
about similar matrices and field extensions
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