Exercise

Let f Z/pZ[X] be an irreducible polynomial of degree n. Suppose that g Z/pZ[X] is a polynomial of degree m for which there is a morphism Z/pZ[X]/(f) -> Z/pZ[X]/(g). Which conclusion about m and n is justified?

n > m

n = m

n m

Yes. Because the morphism is injective, we must have pⁿ = |Z/pZ[X]/(f)| |Z/pZ[X]/(g)| = p^m, which implies n m.