Let \(L/K\) be a finite extension of number fields, unramified at all finite primes, with rings of integers \(O_L, O_K\) and Galois group \(\Gamma\). If \(O_L = O_K[\alpha]\) for some \(\alpha \in O_L\), \(L/K\) is said to have a relative power integral basis (PIB). If \(O_L = O_K[\Gamma](\beta)\) for some \(\beta \in O_L\), \(L/K\) is said to have a relative normal integral basis (NIB). When \(\Gamma\) is cyclic of prime order \(p\) and \(K\) contains a primitive \(p\)th root of unity \(\zeta_p\), the reviewer [Proc. Lond. Math. Soc. (3) 35, 407-422 (1977; Zbl 0374.13002)] showed that if \(L/K\) has a NIB, it has a PIB. The author obtains infinitely many cubic extensions \(L/K\) with \(\zeta_3 \in K\) with a PIB but no NIB, and also infinitely many quadratic extensions \(L/K\) with PIB but no NIB where \(K\) is a real quadratic field. The examples build on earlier results of the author [J. Algebra 235, 104-112 (2001; Zbl 0972.11101)].