Normal integral bases in Kummer extensions of prime degree
Let \(E/F\) be a tamely ramified Kummer extension of prime degree \(p\) of number fields, with Galois group \(G\). The author gives criteria for the ring of integers \(O_ E\) of \(E\) to be a free \((O_ F)G\)-module, i.e. for \(E/F\) to have a normal integral basis. The conditions are related to McCulloh's Stickelberger condition on the Galois module structure of such extensions [cf. \textit{J. Martinet}, ``Algebraic number fields'', Proc. Durham Symp. 1975, 525--538 (1977; Zbl 0359.12014)]. The result is applied in the unramified case to recover a theorem of \textit{F. Kawamoto} [Tokyo J. Math. 8, 275 (1985; Zbl 0577.12003)] and a result of \textit{M. J. Taylor}, \textit{K. Roggenkamp} and \textit{N. P. Byott} [in ``Group rings and class groups'', DMV Semin. 18, 153--210 (1992; Zbl 0811.11068)].
scientific article; zbMATH DE number 703938
scientific article