Unique factorization of monic polynomials

Let R be a commutative integral domain with 1. It is trivial to see that in the polynomial ring R[X], any nonconstant monic polynomial can be factored into a product of nonconstant monic polynomials which are irreducible in R [X]. However, in an arbitrary domain, such a factorization need not be uniq ue. We show uniqueness occurs exactly when R is integrally closed.