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Displaying similar documents to “Generation of class fields by the modular function j 1 , 12 ”

Halfway to a solution of X 2 - D Y 2 = - 3

R. A. Mollin, A. J. Van der Poorten, H. C. Williams (1994)

Journal de théorie des nombres de Bordeaux

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It is well known that the continued fraction expansion of D readily displays the midpoint of the principal cycle of ideals, that is, the point halfway to a solution of x 2 - D y 2 = ± 1 . Here we notice that, analogously, the point halfway to a solution of x 2 - D y 2 = - 3 can be recognised. We explain what is going on.

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.