Paper 1, Section II, F

Groups, Rings and Modules
Part IB, 2009

Prove that a principal ideal domain is a unique factorization domain.

Give, with justification, an example of an element of Z[3]\mathbb{Z}[\sqrt{-3}] which does not have a unique factorization as a product of irreducibles. Show how Z[3]\mathbb{Z}[\sqrt{-3}] may be embedded as a subring of index 2 in a ring RR (that is, such that the additive quotient group R/Z[3]R / \mathbb{Z}[\sqrt{-3}] has order 2) which is a principal ideal domain. [You should explain why RR is a principal ideal domain, but detailed proofs are not required.]