Discussion
Consider the following statements :
(I) The polynomial ring Z[x] is a principal ideal domain.
(II) The polynomial ring Z[x] is a Euclidean domain.
Which is correct ?
Answer: D
Neither (I) nor (II) is true.
Z[x] is not a principal ideal domain. The ideal \((2, x)\), the polynomials with even constant term, is not principal: a generator would divide both 2 and \(x\), so it would be \(\pm 1\), but \(1 \notin (2, x)\). Another way to see it: \(\mathbf{Z}[x]/(x) \cong \mathbf{Z}\) is not a field, so \((x)\) is prime but not maximal, which cannot happen for a nonzero prime ideal in a PID.
Since every Euclidean domain is a PID, Z[x] is not Euclidean either, so (II) fails as well. What is true is that Z[x] is a unique factorisation domain (Gauss's lemma), which is the property candidates often confuse with being a PID.
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