Some remarks on Hilbert–Speiser and Leopoldt fields of given type

James E. Carter¹

¹ Department of Mathematics College of Charleston 66 George Street Charleston, SC 29424-0001, U.S.A.

Colloquium Mathematicum, Tome 108 (2007) no. 2, pp. 217-223

Résumé

Let $p$ be a rational prime, $G$ a group of order $p$, and $K$ a number field containing a primitive $p$th root of unity. We show that every tamely ramified Galois extension of $K$ with Galois group isomorphic to $G$ has a normal integral basis if and only if for every Galois extension $L/K$ with Galois group isomorphic to $G$, the ring of integers $O_L$ in $L$ is free as a module over the associated order ${\cal A}_{L/K}$. We also give examples, some of which show that this result can still hold without the assumption that $K$ contains a primitive $p$th root of unity.

Détail

DOI : 10.4064/cm108-2-5

Keywords: rational prime group order number field containing primitive pth root unity every tamely ramified galois extension galois group isomorphic nbsp has normal integral basis only every galois extension galois group isomorphic nbsp ring integers nbsp module associated order cal examples which result still without assumption contains primitive pth root unity