2012-33.md

course	course_year	question_number	tags	title	year
Groups, Rings and Modules	IB	33	IB 2012 Groups, Rings and Modules	Paper 2, Section II, G	2012

State Gauss's Lemma. State Eisenstein's irreducibility criterion.

(i) By considering a suitable substitution, show that the polynomial $1+X^{3}+X^{6}$ is irreducible over $\mathbb{Q}$.

(ii) By working in $\mathbb{Z}_{2}[X]$, show that the polynomial $1-X^{2}+X^{5}$ is irreducible over $\mathbb{Q}$.