State Gauss' lemma. State and prove Eisenstein's criterion.

Define the notion of an algebraic integer. Show that if $\alpha$ is an algebraic integer, then $\{f \in \mathbb{Z}[X]: f(\alpha)=0\}$ is a principal ideal generated by a monic, irreducible polynomial.

Let $f=X^{4}+2 X^{3}-3 X^{2}-4 X-11$. Show that $\mathbb{Q}[X] /(f)$ is a field. Show that $\mathbb{Z}[X] /(f)$ is an integral domain, but not a field. Justify your answers.