(i) State Gauss' Lemma on polynomial irreducibility. State and prove Eisenstein's criterion.

(ii) Which of the following polynomials are irreducible over $\mathbb{Q}$ ? Justify your answers.

(a) $x^{7}-3 x^{3}+18 x+12$

(b) $x^{4}-4 x^{3}+11 x^{2}-3 x-5$

(c) $1+x+x^{2}+\ldots+x^{p-1}$ with $p$ prime

[Hint: consider substituting $y=x-1$.]

(d) $x^{n}+p x+p^{2}$ with $p$ prime.

[Hint: show any factor has degree at least two, and consider powers of $p$ dividing coefficients.]