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Canonical number systems can be viewed as natural generalizations of radix representations of ordinary integers to algebraic integers. A slightly modified version of an algorithm of B. Kovács and A. Pethő is presented here for the determination of canonical number systems in orders of algebraic number fields. Using this algorithm canonical number systems of some quartic fields are computed.

Berechnung der Werte von Zetafunktionen totalreeller kubischer Zahlkörper an ganzzahligen Stellen mittels verallgemeinerter Dedekindscher Summen. I.

Ulrich Halbritter (1985)

Journal für die reine und angewandte Mathematik

Binomial squares in pure cubic number fields

Franz Lemmermeyer (2012)

Journal de Théorie des Nombres de Bordeaux

Let $K = ℚ (ω)$ , with $ω^{3} = m$ a positive integer, be a pure cubic number field. We show that the elements $α \in K^{\times}$ whose squares have the form $a - ω$ for rational numbers $a$ form a group isomorphic to the group of rational points on the elliptic curve $E_{m} : y^{2} = x^{3} - m$ . This result will allow us to construct unramified quadratic extensions of pure cubic number fields $K$ .

Capitulation dans certaines extensions non ramifiées de corps quartiques cycliques

Abdelmalek Azizi, Mohammed Talbi (2008)

Archivum Mathematicum

Let $K = k (\sqrt{- p ε \sqrt{l}})$ with $k = ℚ (\sqrt{l})$ where $l$ is a prime number such that $l = 2$ or $l \equiv 5 m o d 8$ , $ε$ the fundamental unit of $k$ , $p$ a prime number such that $p \equiv 1 m o d 4$ and ${(\frac{p}{l})}_{4} = - 1$ , $K_{2}^{(1)}$ the Hilbert $2$ -class field of $K$ , $K_{2}^{(2)}$ the Hilbert $2$ -class field of $K_{2}^{(1)}$ and $G = Gal (K_{2}^{(2)} / K)$ the Galois group of $K_{2}^{(2)} / K$ . According to E. Brown and C. J. Parry [7] and [8], $C_{2, K}$ , the Sylow $2$ -subgroup of the ideal class group of $K$ , is isomorphic to $ℤ / 2 ℤ \times ℤ / 2 ℤ$ , consequently $K_{2}^{(1)} / K$ contains three extensions $F_{i} / K$ $...$

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