Let \(p\) be a prime number, \(\zeta\) a primitive \(p\)th root of unity, \(\pi = \zeta -1\), \(F\) a number field, \(K= F(\zeta )\), \(E_K\) the group of units of (the ring of integers of) \(K\). The reviewer proved [Proc. Lond. Math. Soc. (3) 35, 407–422 (1977; Zbl 0374.13002)] that a cyclic unramified extension \(L/K\) of degree \(p\) has a normal integral basis iff \(L = K(\varepsilon ^{1/p})\) for some unit \(\varepsilon \) of \(K\) satisfying \(\varepsilon \equiv 1 \pmod{\pi ^p}\). If either \([K:F] = 2\) or any prime ideal of \(F\) over \(p\) does not split over \(K\), then the author shows that the map \(N \mapsto NK\) gives a bijection between cyclic unramified extensions \(N/F\) of degree \(p\) with normal integral basis and cyclic unramified extensions \(L/K\) of degree \(p\) with normal integral basis such that \(L = K(\varepsilon ^{1/p})\) is an abelian extension of \(F\), the class \([\varepsilon ]\) of \(\varepsilon \) modulo \(E_K^p\) is not \(= 1\) and the Galois group \(\Delta \) of \(K/F\) acts on \([\varepsilon ]\) via \(\sigma ([\varepsilon ]) = ([\varepsilon ]) ^k\) where \(\sigma (\zeta ) = \zeta ^k\). The reviewer also proved [op. cit.] that for \(K\) as above an unramified cyclic extension of order \(p\) has a power integral basis iff it has a normal integral basis.
The author gives an infinite collection of number fields \(F\) whose degree over \(\mathbb Q\) is a multiple of \(q-1\) where \(q = 2^{2^e} + 1\) is a Fermat prime, such that each such \(F\) has an unramified cyclic extension of degree \(p\) with a normal integral basis but with no power integral basis.