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

Published online by Cambridge University Press:  20 November 2018

Humio Ichimura
Humio Ichimura*
Affiliation:
Faculty of Science, Ibaraki University, Bunkyo 2-1-1, Mito 310-8512, Japan e-mail: hichimur@mx.ibaraki.ac.jp
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Abstract

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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 $F$ .

Keywords

11R33

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 4 , 01 December 2005 , pp. 576 - 579
Copyright
Copyright © Canadian Mathematical Society 2005

References

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