Groups, Rings and Modules | Part IB, 2005

Let RR be the ring of Gaussian integers Z[i]\mathbb{Z}[i], where i2=1i^{2}=-1, which you may assume to be a unique factorization domain. Prove that every prime element of RR divides precisely one positive prime number in Z\mathbb{Z}. List, without proof, the prime elements of RR, up to associates.

Let pp be a prime number in Z\mathbb{Z}. Prove that R/pRR / p R has cardinality p2p^{2}. Prove that R/2RR / 2 R is not a field. If p3mod4p \equiv 3 \bmod 4, show that R/pRR / p R is a field. If p1mod4p \equiv 1 \bmod 4, decide whether R/pRR / p R is a field or not, justifying your answer.

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