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

http://dx.doi.org/10.4153/CMB-2005-052-3
Canad. Math. Bull. 48(2005), 576-579

  Published:2005-12-01
 Printed: Dec 2005

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 $F$.