TY - JOUR

T1 - A note on the normality of unramified, abelian extensions of quadratic extensions

AU - Madden, Daniel J.

AU - Vélez, William Yslas

PY - 1979/12

Y1 - 1979/12

N2 - 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.

AB - 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.

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U2 - 10.1007/BF01301254

DO - 10.1007/BF01301254

M3 - Article

AN - SCOPUS:0041951492

VL - 30

SP - 343

EP - 349

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

SN - 0025-2611

IS - 4

ER -