This note settles, in the negative, two problems about affine Noetherian algebras \(S\) over a field \(k\): (1) Is \(S\) finitely presented? (2) Is \(S \otimes_ k K\) Noetherian for every field extension \(K/k\)? Specifically, it is shown that the following algebra \(S\) is a counterexample to both questions. Assume that \(k\) has positive characteristic and let \(K = k(t_ 1,t_ 2,\dots)\) denote the field of rational functions in countably many variables over \(k\). Define a \(k\)-derivation \(\delta\) of \(K\) by \(\delta(t_ i) = t_{i + 1}\) and let \(S = K[x;\delta]\) be the skew polynomial ring. Then \(S\) is an affine PID (left and right), yet \(S\) is not finitely presented, and \(S \otimes_ k K\) is not Noetherian. In a ``Note added in proof'' the authors remark that Yu. Medvedev (Univ. Ottawa), by modifying their construction, has produced counterexamples to (1) and (2) in all characteristics.