On normal integral bases
Necessary and sufficient conditions for a Galois extension \(K/k\) of number fields to have a normal integral basis (n.i.b.) have been given by \textit{A. Fröhlich} [J. Reine Angew. Math. 209, 39--53 (1962; Zbl 0105.02901)]. A simple necessary condition for existence of a n.i.b. is that \(K/k\) be tamely ramified. If \(k = \mathbb{Q}(\zeta)\) and \(K=\mathbb{Q}(a^{1/\ell},\zeta)\) where \(\zeta\) is a primitive \(\ell\)-th root of unity and \(a\neq \pm 1\) is an \(\ell\)-th power free integer, the extension \(K/k\) is tamely ramified if and only if \(a^{\ell -1}\equiv 1 \bmod \ell^2\). The author gives a system of \(\ell\) congruences modulo \(\ell\), involving units of \(k\), which determine exactly when such extensions \(K/k\) have a normal integral basis. Moreover, an explicit generator for the n.i.b. is given. For example, if \(a\neq \pm 1\) is a square free integer with \(a\equiv \pm 1 \bmod \ell^2\) then a n.i.b. always exists for \(K/k\), and \[ \alpha = \displaystyle \frac1{\ell} \sum^{\ell -1}_{j=0}(-(\varepsilon a)^{1/\ell})^j\] is a generator for the basis where \(\varepsilon =\pm 1\) with \(\varepsilon\) \(\equiv -a \bmod \ell^2\). The author applies the above mentioned results to determine other Kummer extensions for which a n.i.b. exists and can be explicitly stated. Finally, the author shows for \(K=\mathbb{Q}(\sqrt{m},\sqrt{n})\) and \(k=\mathbb{Q}(\sqrt{m})\) with \(m\equiv n\equiv 3 \pmod 4\), \(m<-1\), \(n<0\) and \((m,n)=1\) that \(K/k\) is a tamely ramified extension without a n.i.b.
scientific article; zbMATH DE number 3880826