# Math Help - UFD? Nature of polynomial x in the Ring Q_Z[x]

1. ## UFD? Nature of polynomial x in the Ring Q_Z[x]

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

(a) Show that the only divisors of x in $\mathbb{Q}_\mathbb{Z}[x]$ are the integers (constant polynomials) and first degree polynomials of the form $\frac{1}{n}x$ with $0 \ne n \in \mathbb{Z}$

(b) For each non-zero $n \in \mathbb{Z}$ show that the polynomial $\frac{1}{n}x$ is not irreducible in $\mathbb{Q}_\mathbb{Z}[x]$

(c) Show that x cannot be written as a finite product of irreducible elements in $\mathbb{Q}_\mathbb{Z}[x]$

Would appreciate help with this exercise

Peter

2. ## Re: UFD? Nature of polynomial x in the Ring Q_Z[x]

Hi again,
This continues the problem of your previous posting. What this does is prove the ring in question is not a unique factorization domain. Here's a solution: