Normal bases for quadratic extensions inside cyclotomic fields (Q1924590)

Let \(L= \mathbb{Q}[\zeta_p]\) be the cyclotomic extension, \(p\) a prime, and let \(\mathbb{Q}\subset F\subset E\subset L\) with \(E/F\) a quadratic extension. The author constructs an explicit normal basis of \({\mathcal O}_E\) over \({\mathcal O}_F\). As a consequence, he obtains a cyclic extension \(L/F\) of degree 4 whose intermediate layers have normal integral bases but such that \(L/F\) does not. This result complements results of \textit{J. Brinkhuis} [J. Reine Angew. Math. 375/376, 157-166 (1987; Zbl 0609.12009)] and \textit{C. Greither} [J. Number Theory 35, 180-193 (1990; Zbl 0718.11053)].