manuscripta math~ 30, 343- 349 (1980) manuscripta mathematica ~) by Sl~i~l[er-Verla s 1980 A NOTE ON THE NORMALITY OF UNRAMIFIED, ABELIAN EXTENSIONS OF QUADRATIC EXTENSIONS Daniel J. Madden and William Yslas Velez Let F, K and L be algebraic number flelds such that F C K C L, [K:F] = 2 and [L:K] = n. It is a sim- ple consequence of the class field theory that, if L is an abellan, unramlfled 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 con- structing for any field F with even class number a tower of fields F C KC L with [K:F] - 2, [L:K] = 2 where L is unramified--over K, but L is not normal over F. INTRODUCTION Let F and K be algebraic number fields with F C K; it is well-known that certain unramified ex- tensions of K are necessarily normal over F, most notably the Hilbert class field and the genus fleld of K. This givea rise to the general question: Under what conditions is an unramified extension L of K a normal extension of the subfield u This question has been studied by Hesse [2] and more recently by Gogia and Luthar [i]. As a simple consequence of the class field theory we note that, if [K:F] - 2, [L:K] - n, L is an abe- lian, unramified extension of K, and (n,h) - i, where h is the class number of F, then L is nor- me1 over F. This generalizes the results of [1]. OO25-2611/80/OO30/O343/$O1.40 343