Displaying similar documents to “On the diophantine equation y 2 + D m = p n ”

The conductor of a cyclic quartic field using Gauss sums

Blair K. Spearman, Kenneth S. Williams (1997)

Czechoslovak Mathematical Journal

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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.