Normal integral bases of \(\infty\)-ramified abelian extensions of totally real number fields.
Let \(F\) be a totally real algebraic number field, \(\mathfrak l\) a product of (some) real primes of \(F\) and \(K\) an abelian extension of \(F\) with conductor \(\mathfrak l\). \textit{J. Brinkhuis} [Bull. Lond. Math. Soc. 24, 236--242 (1992; Zbl 0768.11045)] proved that if \(K/F\) has a normal integral basis then \(\text{Gal}(K/F)\) must be an elementary 2-group. In the present paper, the authors construct a maximal extension field \(L^{\mathfrak l}\) of \(F\) such that all abelian extensions \(K\) of \(F\) with conductor dividing \(\mathfrak l\) and having a normal integral basis over \(F\) are exactly the fields between \(F\) and \(L^{\mathfrak l}\). The proof uses Kummer theory and a result of \textit{L. N. Childs} [Proc. Lond. Math. Soc. (3) 35, 407--422 (1977; Zbl 0374.13002)] to characterize the existence of normal integral bases for quadratic extensions unramified outside \(\infty\). The larger part of the paper gives a detailed and explicit description of the situation where \(F\) is either a real quadratic or a cyclic cubic number field.
scientific article; zbMATH DE number 1868556
scientific article