Characteristic polynomial splits into linear factors

[Hello Kitty]

Is it true that the characteristic polynomial of an n by n matrix over GF(q) splits into linear factors over GF(q^n)?

I see that it must do if the polynomial is irreducible but what if it isn't?

 

[morphism]

Let p be an irreducible polynomial of degree k over GF(q). What does the splitting field of p (over GF(q)) look like?