Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

The computation of a certain metric invariant of an algebraic number field
HTML articles powered by AMS MathViewer

by Horst Brunotte PDF
Math. Comp. 38 (1982), 627-632 Request permission

Abstract:

Let F be an algebraic number field and denote by $N(a)$ the absolute norm and by $\tilde {a}$ the maximum of the absolute values of the conjugates of the element a of F. Define ${c_F}$ to be the best possible constant with the property: For every $a \in F$ there exists a unit u of F such that $\widetilde {ua} \leqslant {c_F}N{(a)^{1/[F:{\mathbf {Q}}]}}$. An algorithm for the computation of ${c_F}$ is described and some examples are given.
References
    W. E. H. Berwick, "Algebraic number fields with two independent units," Proc. London Math. Soc., v. 34, 1932, pp. 360-378. S. I. Borewicz & I. R. Šafarevič, Zahlentheorie, Birkhäuser, Basel-Stuttgart, 1966.
  • Horst Brunotte, Bemerkungen zu einer metrischen Invarianten algebraischer Zahlkörper, Monatsh. Math. 90 (1980), no. 3, 171–184 (German, with English summary). MR 596884, DOI 10.1007/BF01295362
  • Helmut Hasse, Arithmetische Bestimmung von Grundeinheit und Klassenzahl in zyklischen kubischen und biquadratischen Zahlkörpern, Abh. Deutsch. Akad. Wiss. Berlin. Math.-Nat. Kl. 1948 (1948), no. 2, 95 pp. (1950) (German). MR 33863
  • Helmut Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952 (German). MR 0049239
  • Helmut Hasse, Zahlentheorie, Akademie-Verlag, Berlin, 1969 (German). Dritte berichtigte Auflage. MR 0253972
  • Otto Körner, Erweiterter Goldbach-Vinogradovscher Satz in beliebigen algebraischen Zahlkörpern, Math. Ann. 143 (1961), 344–378 (German). MR 123552, DOI 10.1007/BF01470615
  • Donald E. Knuth, The art of computer programming. Vol. 2: Seminumerical algorithms, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0286318
  • G. J. Rieger, Über die Darstellung ganzer algebraischer Zahlen durch Quadrate, Arch. Math. (Basel) 14 (1963), 22–28 (German). MR 151444, DOI 10.1007/BF01234915
  • Carl Siegel, Darstellung total positiver Zahlen durch Quadrate, Math. Z. 11 (1921), no. 3-4, 246–275 (German). MR 1544496, DOI 10.1007/BF01203627
  • R. Smadja, Calculs Effectifs sur les Idéaux des Corps de Nombres Algébriques, Univ. D’Aix-Marseille, U.E.R. Sci. de Luminy, 1976.
  • Emery Thomas, Fundamental units for orders in certain cubic number fields, J. Reine Angew. Math. 310 (1979), 33–55. MR 546663
  • B. L. van der Waerden, "Ein logarithmenfreier Beweis des Dirichletschen Einheitensatzes," Abh. Math. Sem. Univ. Hamburg, v. 6, 1928, pp. 259-262.
Similar Articles
Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Math. Comp. 38 (1982), 627-632
  • MSC: Primary 12A99; Secondary 12-04, 12A45
  • DOI: https://doi.org/10.1090/S0025-5718-1982-0645677-8
  • MathSciNet review: 645677