2016-33.md

course	course_year	question_number	tags	title	year
Groups, Rings and Modules	IB	33	IB 2016 Groups, Rings and Modules	Paper 3, Section II, E	2016

(a) Define what is meant by an algebraic integer $\alpha$. Show that the ideal

$$I={h \in \mathbb{Z}[x] \mid h(\alpha)=0}$$

in $\mathbb{Z}[x]$ is generated by a monic irreducible polynomial $f$. Show that $\mathbb{Z}[\alpha]$, considered as a $\mathbb{Z}$-module, is freely generated by $n$ elements where $n=\operatorname{deg} f$.

(b) Assume $\alpha \in \mathbb{C}$ satisfies $\alpha^{5}+2 \alpha+2=0$. Is it true that the ideal (5) in $\mathbb{Z}[\alpha]$ is a prime ideal? Is there a ring homomorphism $\mathbb{Z}[\alpha] \rightarrow \mathbb{Z}[\sqrt{-1}]$ ? Justify your answers.

(c) Show that the only unit elements of $\mathbb{Z}[\sqrt{-5}]$ are 1 and $-1$. Show that $\mathbb{Z}[\sqrt{-5}]$ is not a UFD.