This is very confusing, you should pay more attention to your typing. First you assert that " So you know $\chi_a = m_a p$ " . What is " $m_a p$ " ?? If it is $ m_a$, the minimal polynomial of the algebraic element a over F (you sometimes denoted it also by $p_a$ !), then I agreed that that $m_a = q_a$ (which is the minimal polynomial of the endo. L_a), and so $m_a$ divides the characteristic polynomial $\chi_a$ (defined in your def.1) because of C-H...

But where does the claimed equality come from ? Surely, in all generality, if an irreducible pol. divides another pol., they have no reason to coincide // Then you say that $m_a$ and $\chi_a$ share the same roots. Why ? In your def.1, $chi_a$ is just a determinant, you know nothing of its roots. Did you make a confusion with def. 2 ?