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

Let F, K and L be algebraic number fields such that {Mathematical expression}, [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 {Mathematical expression} with [K:F]=2, [L:K]=2 where L is unramified over K, but L is not normal over F.