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

a) If p(t) is a ploynomial over F, show that p(a) = p(a_1) + ... + p(a_k).
b) Show that \Delta_a(t) = \Delta_{a_1}(t) * ... * \Delta_{a_k}(t)
c) Show that m_a(t) is the LCM of 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 W_i = ker(m_{a_i}(a))

I've got a). I just let 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).