If F is a field, n a positive integer, c(x) in F[x] a polynomial of degree n, and m(x) is in F[x] such that m(x) divides c(x) and if p(x) in F[x] is an irreducible factor of c(x), then p(x) divides m(x),
Is it true that there exists an n*n matrix over F that has characteristic polynomial c(x) and minimal polynomial m(x)?
I was thinking it wouldnt be so if c(x) was not monic, and F[x] is a field of integers. Am I missing something, and how should the proof(or counterexample) go?