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

Published:2005-12-01
Printed: Dec 2005
• Humio Ichimura
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## Abstract

Let $m=p^e$ be a power of a prime number $p$. We say that a number field $F$ satisfies the property $(H_m')$ when for any $a \in F^{\times}$, the cyclic extension $F(\z_m, a^{1/m})/F(\z_m)$ has a normal $p$-integral basis. We prove that $F$ satisfies $(H_m')$ if and only if the natural homomorphism $Cl_F' \to Cl_K'$ is trivial. Here $K=F(\zeta_m)$, and $Cl_F'$ denotes the ideal class group of $F$ with respect to the $p$-integer ring of $F$.
 MSC Classifications: 11R33 - Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10]