1. Consider the ring Z[sqrt(-3)]
(1) Prove 5 is an irreducible element in Z[sqrt(-3)]
(2) Prove 5 is not a prime element in Z[sqrt(-3)]
(3) Is Z[sqrt(-3)] ia UFD?
(4) Is Z[sqrt(-3)]/<5> a field?
2. Prove or disprove
(1) 1+sqrt(-19) is a prime element in Z[1+sqrt(-19)]
(2) Z[1+sqrt(-19)] is UFD
(3) If R is PID, then R[x] is a PID.
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