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

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

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

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

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

Would appreciate help with this exercise

Peter

1 Attachment(s)

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:

Attachment 28399