Relative integral bases of abelian \(q\)-fields
The author considers the question of existence of a relative integral basis for abelian \(q\)-fields. For abelian \(q\)-fields \(K\) and \(L\) with \(K\subset L\) and a rational prime \(p\), let \(e(p,K)\) and \(e(p,L)\) denote the ramification indices of \(p\) in \(K\) and \(L\), respectively. If \(q\neq 2\) and \(e(p,L)\leq [L:K]\) for all primes \(p\) with \(e(p,L)\neq e(p,K)\), then it is shown that \(L/K\) has a relative integral basis. When \(q=2\), some additional hypothesis on the ramification of 2 is needed to obtain a similar result. The proofs of the results use the well-known criteria of \textit{E. Artin} [Colloques Internat. Centre Nat. Rech. Sci. 24, 19-20 (1950; Zbl 0039.02904)] for the existence of relative integral bases.
scientific article; zbMATH DE number 940616