| course |
course_year |
question_number |
tags |
title |
year |
Groups, Rings and Modules |
IB |
35 |
IB |
2017 |
Groups, Rings and Modules |
|
Paper 3, Section II, E |
2017 |
(a) Define what is meant by a Euclidean domain. Show that every Euclidean domain is a principal ideal domain.
(b) Let $p \in \mathbb{Z}$ be a prime number and let $f \in \mathbb{Z}[x]$ be a monic polynomial of positive degree. Show that the quotient ring $\mathbb{Z}[x] /(p, f)$ is finite.
(c) Let $\alpha \in \mathbb{Z}[\sqrt{-1}]$ and let $P$ be a non-zero prime ideal of $\mathbb{Z}[\alpha]$. Show that the quotient $\mathbb{Z}[\alpha] / P$ is a finite ring.