Prove that the rational form of a matrix remains the same over subfield

Question

Let F be a subfield of the complex number. How to show that the rational form of a matrix over complex is the same over F?

I think we need to use the cyclic decomposition theorem. Finding the rational form depends on cyclic decomposition theorem.

$V = Z(\alpha_1:T) \bigoplus ... \bigoplus Z(\alpha_r:T) $ Where $p_{i+1}|p_i$ and $p_i$ is the annihilator of $\alpha_i$.This representation is unique

Now I am stuck at the fact that the dimension of V changes when viewed over the field and over the subfield.Then how can we claim that this representation is unique?