## The ring of polynomials over a field with no linear term is not a UFD

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.

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## Comments

I think the 4th line should be \alpha\beta= a_0b_0+x^2(a_0b(x) + b_0a(x) +x^2a(x)b(x))\in R

Thanks!