diagonalising matrices over different fields Say I have two fields $F$ and $F'$ such that $F\subseteq F'$. Say for a matrix $A \in M_{n}(F)$ is diagonalisable over $F$, then does it automatically mean that $A$ is diagonalisable over $F'$?
The question I was dealing was a matrix, $A$, say, which I found to be diagonalisable over $\mathbb{R}$ and so I am thinking then that means $A$ is diagonalisable over $\mathbb{C}$ right?

This is correct. If you diagonalize over $F$, then you will be able to diagonalize over $F'\supseteq F$, since you can just use the same operations over $F'$ that you have used over $F$. The converse is not generally true.
