On integral representations by totally positive ternary quadratic forms

Elise Björkholdt

Displaying similar documents to “On integral representations by totally positive ternary quadratic forms”

The size function $h^{0}$ for quadratic number fields

Paolo Francini (2001)

Journal de théorie des nombres de Bordeaux

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We study the quadratic case of a conjecture made by Van der Geer and Schoof about the behaviour of certain functions which are defined over the group of Arakelov divisors of a number field. These functions correspond to the standard function $h^{0}$ for divisors of algebraic curves and we prove that they reach their maximum value for principal Arakelov divisors and nowhere else. Moreover, we consider a function $\tilde{k^{0}}$ , which is an analogue of exp $h^{0}$ defined on the class group, and we show it also...

Unramified quaternion extensions of quadratic number fields

Franz Lemmermeyer (1997)

Journal de théorie des nombres de Bordeaux

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Classical results of Rédei, Reichardt and Scholz show that unramified cyclic quartic extensions of quadratic number fields $k$ correspond to certain factorizations of its discriminant disc $k$ . In this paper we extend their results to unramified quaternion extensions of $k$ which are normal over $ℚ$ , and show how to construct them explicitly.

A quadratic field of prime discriminant requiring three generators for its class group, and related theory

Daniel Shanks, Peter Weinberger (1972)

Acta Arithmetica

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Quadratic integral solutions to double Pell equations

Francesco Veneziano (2011)

Rendiconti del Seminario Matematico della Università di Padova

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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 the 2-primary part of K₂ of rings of integers in certain quadratic number fields

A. Vazzana (1997)

Acta Arithmetica

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1. Introduction. For quadratic fields whose discriminant has few prime divisors, there are explicit formulas for the 4-rank of $K ₂_{E}$ . For quadratic fields whose discriminant has arbitrarily many prime divisors, the formulas are less explicit. In this paper we will study fields of the form $ℚ (\sqrt (p ₁ . . . p_{k}))$ , where the primes $p_{i}$ are all congruent to 1 mod 8. We will prove a theorem conjectured by Conner and Hurrelbrink which examines under what conditions the 4-rank of $K ₂_{E}$ is zero for such fields. In the course...

On determining the quadratic subfields of $Z_{2}$ -extensions of complex quadratic fields

Joseph E. Carroll (1975)

Compositio Mathematica

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Integral solutions of the equation in the quadratic realms of rationality $ξ^{2} + η^{2} = ζ^{2}$

Harris Hancock (1921)

Journal de Mathématiques Pures et Appliquées

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Maximal unramified extensions of imaginary quadratic number fields of small conductors, II

Ken Yamamura (2001)

Journal de théorie des nombres de Bordeaux

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In the previous paper [15], we determined the structure of the Galois groups $Gal (K_{u r} / K)$ of the maximal unramified extensions $K_{u r}$ of imaginary quadratic number fields $K$ of conductors $≦ 1000$ under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors $≧ 723$ ) and give a table of $Gal (K_{u r} / K)$ . We update the table (under GRH). For 19 exceptional fields $K$ of them, we determine $Gal (K_{u r} / K)$ . In particular, for $K = 𝐐 (\sqrt{- 856})$ , we obtain $Gal (K_{u r} / K) ≅ \tilde{S_{4}} \times C_{5} and K_{u r} = K_{4}$ , the fourth Hilbert class field of $K$ . This is the first example of a number...

Maximal unramified extensions of imaginary quadratic number fields of small conductors

Ken Yamamura (1997)

Journal de théorie des nombres de Bordeaux

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We determine the structures of the Galois groups Gal $(K_{u r} / K)$ of the maximal unramified extensions $K_{u r}$ of imaginary quadratic number fields $K$ of conductors $≦ 420 (≦ 719$ under the Generalized Riemann Hypothesis). For all such $K$ , $K_{u r}$ is $K$ , the Hilbert class field of $K$ , the second Hilbert class field of $K$ , or the third Hilbert class field of $K$ . The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is essential. We...

Euclid's algorithm in complex quartic fields

Richard Lakein (1972)

Acta Arithmetica

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On an approximation property of Pisot numbers II

Toufik Zaïmi (2004)

Journal de Théorie des Nombres de Bordeaux

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Let $q$ be a complex number, $m$ be a positive rational integer and $l_{m} (q) = inf {|P (q)|, P \in ℤ_{m} [X], P (q) \neq 0}$ , where $ℤ_{m} [X]$ denotes the set of polynomials with rational integer coefficients of absolute value $\leq m$ . We determine in this note the maximum of the quantities $l_{m} (q)$ when $q$ runs through the interval $] m, m + 1 [$ . We also show that if $q$ is a non-real number of modulus $> 1$ , then $q$ is a complex Pisot number if and only if $l_{m} (q) > 0$ for all $m$ .