On a Theorem of Kawamoto on Normal Bases of Rings of Integers, II

Ichimura, Humio

Canadian mathematical bulletin, Tome 48 (2005) no. 4, pp. 576-579

Résumé

Let $m={{p}^{e}}$ be a power of a prime number $p$ . We say that a number field $F$ satisfies the property $\left( {{H}^{'}}_{m} \right)$ when for any $a\in {{F}^{\times }}$ , the cyclic extension $F\left( {{\zeta }_{m}},{{a}^{1/m}} \right)/F\left( {{\zeta }_{m}} \right)$ has a normal $p$ -integral basis. We prove that $F$ satisfies $\left( {{H}^{'}}_{m} \right)$ if and only if the natural homomorphism $C{{l}^{'}}_{F}\to C{{l}^{'}}_{K}$ is trivial. Here $K=F\left( {{\zeta }_{m}} \right)$ , and $C{{l}^{'}}_{F}$ denotes the ideal class group of $F$ with respect to the $p$ -integer ring of $F$ .

Détail

DOI : 10.4153/CMB-2005-052-3

Mots-clés : 11R33