A note on the normality of unramified, abelian extensions of quadratic extensions

Abstract

Let F, K and L be algebraic number fields such that$F \subseteq K \subseteq L$, [K∶F]=2 and [L∶K]=n. It is a simple consequence of the class field theory that, if L is an abelian, unramified extension of K and (n,h)=1, where h is the class number of F, then L is normal over F. The purpose of this note is to point out the necessity of the condition (n,h)=1 by constructing for any field F with even class number a tower of fields$F \subseteq K \subseteq L$ with [K∶F]=2, [L∶K]=2 where L is unramified over K, but L is not normal over F.