Note on Galois descent of a normal integral basis of acyclic extension of degree \(p\) (Q2268708)

Let \(p\) be an odd prime, \(F\) an algebraic number field which is unramified at \(p\), and \(K = F(\zeta_p)\). In this note it is shown that for any tame Galois extension \(N/F\) with \([N:F]=p\) one has: \(N/F\) has a normal integral basis if and only if \(NK/K\) has a normal integral basis.