Let be a field. Prove that the subset consisting of all polynomials whose linear coefficient is zero is a subring. Prove also that is not a unique factorization domain.
To show that is a subring, we need to see that it is closed under subtraction and multiplication. To that end, let and be in . Then and . So is a subring.
Next, we claim that and are irreducible in . To see this, note that no element of has degree 1. If factors in , then computing the degree of both sides we have . Since neither of these degrees is 1, one must be 0, so that without loss of generality, is constant and thus a unit. So is irreducible. Similarly, is irreducible because if two nonnegative integers sum to 3 then one of them must be 0 or 1. So and are irreducible in .
Now , so that has at least two distinct irreducible factorizations in . Thus is not a unique factorization domain.