Suppose that is a finite dimensional vector space over . Let for some liner .
Then:
a) If is a ploynomial over F, show that .
b) Show that
c) Show that is the LCM of and thus a product of them if they are pairwise coprime.
d) Suppose they are pairwise coprime. Show that
I've got a). I just let 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).