Paper 1, Section II, F
Part IB, 2009
Prove that a principal ideal domain is a unique factorization domain.
Give, with justification, an example of an element of which does not have a unique factorization as a product of irreducibles. Show how may be embedded as a subring of index 2 in a ring (that is, such that the additive quotient group has order 2) which is a principal ideal domain. [You should explain why is a principal ideal domain, but detailed proofs are not required.]