| course |
course_year |
question_number |
tags |
title |
year |
Groups, Rings and Modules |
IB |
32 |
IB |
2009 |
Groups, Rings and Modules |
|
Paper 1, Section II, F |
2009 |
Prove that a principal ideal domain is a unique factorization domain.
Give, with justification, an example of an element of $\mathbb{Z}[\sqrt{-3}]$ which does not have a unique factorization as a product of irreducibles. Show how $\mathbb{Z}[\sqrt{-3}]$ may be embedded as a subring of index 2 in a ring $R$ (that is, such that the additive quotient group $R / \mathbb{Z}[\sqrt{-3}]$ has order 2) which is a principal ideal domain. [You should explain why $R$ is a principal ideal domain, but detailed proofs are not required.]