On normal bases of some ring extensions in number fields. I
This paper studies the existence of normal bases for rings of \(S\)-integers of a Galois extension \(K/k\) of a number field \(k\). The tool used is a certain ideal resolvent constructed by using sums \(\sum\chi(s^{-1})s(\alpha)\) on the elements of the Galois group for each character \(\chi\) already introduced in \textit{F. Kawamoto} and \textit{K. Komatsu} [J. Algebra 163, 335-347 (1994; Zbl 0810.11064)]. After studying criteria of non-decomposition of prime ideals in relative cyclotomic extensions, the author gives a sufficient condition for non-existence of normal bases (sec. 4). Finally, the main theorem of sec. 5 discusses the cases of subextensions of the cyclotomic field \(\mathbb{Q}(\zeta_p)\), with \(p\) a prime number and gives the precise list of the four cases where the ring of integers of \(K\) does have a normal base over the one of \(k\).
scientific article; zbMATH DE number 927038