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The conductor of a cyclic quartic field using Gauss sums

Blair K. SpearmanKenneth S. Williams — 1997

Czechoslovak Mathematical Journal

Let Q denote the field of rational numbers. Let K be a cyclic quartic extension of Q . It is known that there are unique integers A , B , C , D such that K = Q A ( D + B D ) , where A is squarefree and odd , D = B 2 + C 2 is squarefree , B > 0 , C > 0 , G C D ( A , D ) = 1 . The conductor f ( K ) of K is f ( K ) = 2 l | A | D , where l = 3 , if D 2 ( mod 4 ) or D 1 ( mod 4 ) , B 1 ( mod 2 ) , 2 , if D 1 ( mod 4 ) , B 0 ( mod 2 ) , A + B 3 ( mod 4 ) , 0 , if D 1 ( mod 4 ) , B 0 ( mod 2 ) , A + B 1 ( mod 4 ) . A simple proof of this formula for f ( K ) is given, which uses the basic properties of quartic Gauss sums.

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