Weber's class invariants revisited

Reinhard Schertz

Displaying similar documents to “Weber's class invariants revisited”

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

Generation of class fields by the modular function $j_{1, 12}$

Kuk Jin Hong, Ja Kyung Koo (2000)

Acta Arithmetica

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On the diophantine equation $y^{2} + D^{m} = p^{n}$

Masao Toyoizumi (1983)

Acta Arithmetica

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The diophantine equation $a x^{2} + b x y + c y^{2} = N, D = b^{2} - 4 a c > 0$

Keith Matthews (2002)

Journal de théorie des nombres de Bordeaux

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We make more accessible a neglected simple continued fraction based algorithm due to Lagrange, for deciding the solubility of $a x^{2} + b x y + c y^{2} = N$ in relatively prime integers $x, y$ , where $N \neq 0$ , gcd $(a, b, c) = gcd (a, N) = 1 et D = b^{2} - 4 a c > 0$ is not a perfect square. In the case of solubility, solutions with least positive y, from each equivalence class, are also constructed. Our paper is a generalisation of an earlier paper by the author on the equation $x^{2} - D y^{2} = N$ . As in that paper, we use a lemma on unimodular matrices that gives a much simpler proof than Lagrange’s...

On some equations over finite fields

Ioulia Baoulina (2005)

Journal de Théorie des Nombres de Bordeaux

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In this paper, following L. Carlitz we consider some special equations of $n$ variables over the finite field of $q$ elements. We obtain explicit formulas for the number of solutions of these equations, under a certain restriction on $n$ and $q$ .

On Taylor's conjecture for Kummer orders

Philippe Cassou-Noguès, Anupam Srivastav (1990)

Journal de théorie des nombres de Bordeaux

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On some special quartic reciprocity laws

Emma Lehmer (1972)

Acta Arithmetica

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Representation of prime powers in arithmetical progressions by binary quadratic forms

Franz Halter-Koch (2003)

Journal de théorie des nombres de Bordeaux

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Let $Γ$ be a set of binary quadratic forms of the same discriminant, $Δ$ a set of arithmetical progressions and $m$ a positive integer. We investigate the representability of prime powers $p^{m}$ lying in some progression from $Δ$ by some form from $Γ$ .

On integral representations by totally positive ternary quadratic forms

Elise Björkholdt (2000)

Journal de théorie des nombres de Bordeaux

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Let $K$ be a totally real algebraic number field whose ring of integers $R$ is a principal ideal domain. Let $f (x_{1}, x_{2}, x_{3})$ be a totally definite ternary quadratic form with coefficients in $R$ . We shall study representations of totally positive elements $N \in R$ by $f$ . We prove a quantitative formula relating the number of representations of $N$ by different classes in the genus of $f$ to the class number of $R [\sqrt{- c_{f} N}]$ , where $c_{f} \in R$ is a constant depending only on $f$ . We give an algebraic proof of a classical result of H. Maass...