Relative Galois module structure of integers of abelian fields
The paper explicitly describes the associated order \({\mathfrak A}_{L/K}\) of a number field extension \(L/K\) with \(L/\mathbb Q\) abelian and \(K\) a cyclotomic field. As a first corollary it is shown that \({\mathfrak A}_{L/K}\) is isomorphic to the ring of integers in \(L\). Special cases have been studied before by \textit{H. W. Leopoldt} [J. Reine Angew. Math. 201, 119--149 (1959; Zbl 0098.03403)] where \(K=\mathbb Q\), by \textit{S.-P. Chan} and \textit{C.-H. Lin} [J. Reine Angew. Math. 434, 205--220 (1993; Zbl 0753.11038)] where \(L\) is cyclotomic, and by \textit{W. Bley} [Can. Math. Bull. 38, 141--148 (1995; Zbl 0830.11037)] where \(\mathbb Q(\zeta_n)/L\) is tame and \(n\) is the conductor of \(L\). A second corollary says that, for more general subfields \(K\) of \(L\), \({\mathfrak A}_{L/K}\) is the maximal order in \(K[\text{Gal}(L/L_m)]\) provided that \(L/K\) is cyclic, totally wildly ramified, and disjoint to \(\mathbb Q(\zeta_{m'})/K\). Here, \(m'=\) conductor of \(K\), \(m=m' \cdot \prod p\) with \(p\) running through the prime divisors of \(n\) which do not divide \(m'\), and \(L=L\cap \mathbb Q(\root m\of {1})\).
scientific article; zbMATH DE number 912502
scientific article