# Thread: Failure of Unique Factorization

1. ## Failure of Unique Factorization

Let $\mathbb{Q}_\mathbb{Z}[x]$ denote the set of polynomials with rational coefficients and integer constant terms.

Prove that the only two units in $\mathbb{Q}_\mathbb{Z}[x]$ are 1 and -1.

Help with this exercise would be appreciated.

Peter

2. ## Re: Failure of Unique Factorization

Hi,
Regarding (a) I think the solution is as follows:

We need to show the p is irreducible in $\mathbb{Q}_\mathbb{Z}[x]$

That is if p = ab for p, a, b $\in \mathbb{Q}_\mathbb{Z}[x]$ then at least one of a or b must be a unit

The above is exactly what you must prove for your second post. First what are the units in your ring? I don't believe your "proof". Attached is a correct discussion: