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


Published:20051201
Printed: Dec 2005
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 $F$.