Chapter 8.1, Problem 7E

Prove that

${\rm I}\left[x\right]=\left\{a_0+a_{1}x+\mathrm{...}+a_n{x}^n|a_0=2kfork\in \mathbb{Z}\right\}$, the set of all polynomials in $\mathbb{Z}\left[x\right]$ with even constant term, is an ideal of $\mathbb{Z}\left[x\right]$.

Show that ${\rm I}\left[x\right]$ is not a principal ideal; that is, show that there is no $f\left(x\right)\in \mathbb{Z}\left[x\right]$ such that

${\rm I}\left[x\right]=\left(f\left(x\right)\right)=\left\{f\left(x\right)g\left(x\right)|g\left(x\right)\in \mathbb{Z}\left[x\right]\right\}.$

Show that ${\rm I}\left[x\right]$ is an ideal generated by two elements in $\mathbb{Z}\left[x\right]$ that is,

${\rm I}\left[x\right]=\left(x,2\right)=\left\{xf\left(x\right)+2g\left(x\right)|f\left(x\right),g\left(x\right)\in \mathbb{Z}\left[x\right]\right\}$.