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

Pierre Kaplan; Kenneth Williams

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A note on the class number of real quadratic fields

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On the parity of the class number of the field $ℚ (\sqrt{p}, \sqrt{q}, \sqrt{r})$

Michal Bulant (1998)

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A rational sixteenth power reciprocity law

Philip Leonard, Kenneth Williams (1977)

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The conductor of a cyclic quartic field using Gauss sums

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

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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 $K = Q (\sqrt{A (D + B \sqrt{D})}),$ 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.

Displaying similar documents to “On the class numbers of Q(√±2p) modulo 16, for p ≡ 1 (mod 8) a prime”

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