Suppose that $\displaystyle V = W_1 + ... + W_k$ is a finite dimensional vector space over $\displaystyle F$. Let $\displaystyle a = a_1 + ... + a_k$ for some liner $\displaystyle a_i : W_i \rarr W_i$.

a) If $\displaystyle p(t)$ is a ploynomial over F, show that $\displaystyle p(a) = p(a_1) + ... + p(a_k)$.
b) Show that $\displaystyle \Delta_a(t) = \Delta_{a_1}(t) * ... * \Delta_{a_k}(t)$
c) Show that $\displaystyle m_a(t)$ is the LCM of $\displaystyle m_1(t),...,m_k(t)$ and thus a product of them if they are pairwise coprime.
d) Suppose they are pairwise coprime. Show that $\displaystyle W_i = ker(m_{a_i}(a))$

I've got a). I just let $\displaystyle p(t)$ be a general poly, and kind of multiplies everything out, rearranged a bit.

I think c) follows almost directly from b), but I don't know how to show b) or d).