Since the minimal polynomial of T splits in F and since the invariant factors of T divide the minimal polynomial, it follows that each invariant factor splits in F. So since the characteristic polynomial is given by the product of the invariant factors, the characteristic polynomial of T splits into linear factors over F. Hence, all eigenvalues of T exist in the base field F. So the Jordan Canonical form applies and gives a basis for V relative to which the matrix for T is upper triangular.