Bioctic Gauss sums and sixteenth power residue difference sets

Ronald Evans

Displaying similar documents to “Bioctic Gauss sums and sixteenth power residue difference sets”

A rational sixteenth power reciprocity law

Philip Leonard, Kenneth Williams (1977)

Acta Arithmetica

Similarity:

On the class numbers of Q(√±2p) modulo 16, for p ≡ 1 (mod 8) a prime

Pierre Kaplan, Kenneth Williams (1982)

Acta Arithmetica

Similarity:

The conductor of a cyclic quartic field using Gauss sums

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

Czechoslovak Mathematical Journal

Similarity:

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 $K = Q (\sqrt{A (D + B \sqrt{D})}),$ 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 $l = \{\begin{matrix} 3, if D \equiv 2 (mod 4) or D \equiv 1 (mod 4), B \equiv 1 (mod 2), \\ 2, if D \equiv 1 (mod 4), B \equiv 0 (mod 2), A + B \equiv 3 (mod 4), \\ 0, if D \equiv 1 (mod 4), B \equiv 0 (mod 2), A + B \equiv 1 (mod 4) . \end{matrix}$ A simple proof of this formula for $f (K)$ is given, which uses the basic properties of quartic Gauss sums.