Nothing Special   »   [go: up one dir, main page]
Skip to main content

Advertisement

Springer Nature Link
Log in

Some computational aspects of, and the use of computers in, algebraic number theory

Einige rechnerische Aspekte und die Anwendung von Rechenanlagen in der algebraischen Zahlentheorie

  • Published:
Computing Aims and scope Submit manuscript

Summary

We discuss here some basic problems of algebraic number theory under a computational point of view. In particular, an explanation is given of how the application of a computer has contributed to the progress of research in this area. Several number-theoretic questions are expounded for which the use of a computer seems to be appropriate but has (apparently) not extensively been made.

Zusammenfassung

Wir diskutieren hier einige grundsätzliche Probleme der algebraischen Zahlentheorie unter rechnrischen Gesichtspunkten. Insbesondere wird ausgeführt, wie die Anwendung von Rechenanlagen zum Fortschritt der Forschung auf diesem Gebiet beigetragen hat. Verschiedene zahlentheoretische Fragen werden angeschnitten, für welche die Anwendung von Rechenanlagen angemessen zu sein scheint, aber (offenbar) noch nicht intensiv praktiziert worden ist.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Berwick, W. E. H.: Integral Bases. Cambridge Tracts in Math. and Math. Phys.22 (1927).

  2. Billevič, K. K.: On units of algebraic fields of third and fourth degree. Mat. Sb. N. S.40 (82), 123–136 (1956).

    Google Scholar 

  3. Billevič, K. K.: Letter to the editors. Mat. Sb. N. S.48 (90), 256 (1959).

    Google Scholar 

  4. Billevič, K. K.: A theorem on unit elements of algebraic fields of ordern. Mat. Sb. N. S.64 (106), 145–152 (1964).

    Google Scholar 

  5. Birch, B. J.: Conjectures concerning elliptic curves. Proc. Sympos. Pure Math. VIII, 106–112. Amer. Math. Soc., Providence, R. I., 1965.

    Google Scholar 

  6. Birch, B. J., andH. P. F. Swinnerton-Dyer: Notes on elliptic curves I, II. J. Reine Angew. Math.212, 7–25 (1963);218, 79–108 (1965).

    Google Scholar 

  7. Bullig, G.: Ein periodischesVerfahren zur Berechnung eines Systems von Grundeinheiten in den total reellen kubischen Körpern. Abh. Math. Sem. Univ. Hamburg12, 369–414 (1938).

    Google Scholar 

  8. Cantor, D. G., P. H. Galyean, andH. G. Zimmer: A continued fraction algorithm for real algebraic numbers. To appear.

  9. Cassels, J. W. S.: The rational solutions of theDiophantine equationY 2=X3−D. Acta Math.82, 243–273 (1950);84, 299 (1951).

    Google Scholar 

  10. Cassels, J. W. S., andM. J. T. Guy: On theHasse principle for cubic surfaces. Mathematika13, 111–120 (1966).

    Google Scholar 

  11. Churchhouse, R. F., andS. T. E. Muir: Continued fractions, algebraic numbers and modular invariants. J. Inst. Math. Appl.5, 318–328 (1969).

    Google Scholar 

  12. Cockayne, E. J.: Computation ofGalois group elements of a polynomial equation. Math. Comp.23, 425–428 (1969).

    Google Scholar 

  13. Cohn, H.: Periodic algorithm for cubic forms I, II. Amer. J. Math.74, 821–833 (1952);76, 904–914 (1954).

    Google Scholar 

  14. Cohn, H.: Numerical study of signature rank of cubic cyclotomic units. Math. Tables Aids Comp.8, 186–188 (1954).

    Google Scholar 

  15. Cohn, H.: A device for generating fields of even class number. Proc. Amer. Math. Soc.7, 595–598 (1956).

    Google Scholar 

  16. Cohn, H.: A numerical study ofDedekind's cubic class number formula. J. Res. Nat. Bur. Standards59, 265–271 (1957).

    Google Scholar 

  17. Cohn, H.: A computation of some bi-quadratic class numbers. Math. Comp.12, 213–217 (1958).

    Google Scholar 

  18. Cohn, H.: A numerical study ofWeber's real class number calculation I. Numer. Math.2, 347–362 (1960).

    Google Scholar 

  19. Cohn, H.: Calculation of class numbers by decomposition into three integral squares in the fields of 21/2 and 31/2. Amer. J. Math.83, 33–56 (1961).

    Google Scholar 

  20. Cohn, H.: Proof thatWeber's normal units are not perfect powers. Proc. Amer. Math. Soc.12, 964–966 (1961).

    Google Scholar 

  21. Cohn, H.: A numerical study of the relative class numbers of real quadratic integral domains. Math. Comp.16, 127–140 (1962).

    Google Scholar 

  22. Cohn, H.: A numerical study of units in composite real quartic and octic fields. Proc. Atlas Sympos. No. 2, Oxford, England, 1969 (to appear).

  23. Cohn, H. andS. Gorn: A computation of cyclic cubic units. J. Res. Nat. Bur. Standards59, 155–168 (1957).

    Google Scholar 

  24. Dade, E. C., O. Taussky, andH. Zassenhaus: On the semigroup of ideal classes in an order of an algebraic number field. Bull. Amer. Math. Soc.67, 305–308 (1961).

    Google Scholar 

  25. Dade, E. C., O. Taussky, andH. Zassenhaus: On the theory of orders, in particular on the semigroup of ideal classes and genera of an order in an algebraic number field. Math. Ann.148, 31–64 (1962).

    Google Scholar 

  26. Frank, E.: Computer use in continued fraction expansions. Math. Comp.23, 429–435 (1969).

    Google Scholar 

  27. Godwin, H. J.: The determination of units in totally real cubic fields. Proc. Cambridge Philos. Soc.56, 318–321 (1960).

    Google Scholar 

  28. Godwin, H. J.: The determination of class-numbers of totally real cubic fields. Proc. Cambridge Philos. Soc.57, 728–730 (1961).

    Google Scholar 

  29. Godwin, H. J.: Computations relating to cubic fields. Proc. Altas Sympos. No. 2, Oxford, England, 1969 (to appear).

  30. Hasse, H.: Arithmetische Bestimmung von Grundeinheit und Klassenzahl in zyklischen kubischen und biquadratischen Zahlkörpern. Abh. Deutsch. Akad. Wiss. Berlin, Math.-Nat. Kl. (1948)2, 95 pp. (1950).

  31. Hasse, H.: Über die Klassenzahl abelscher Zahlkörper. Berlin: Akademie-Verlag, 1952.

    Google Scholar 

  32. Hasse, H.: Über den Klassenkörper zum quadratischen Zahlkörpper mit der Diskriminante −47. Acta Arith.9, 419–434 (1964).

    Google Scholar 

  33. Hasse, H.: Über die Klassenzahl des Körpers mit einer Primzahlp≠2. J. Number Theory1, 231–234 (1969).

    Google Scholar 

  34. Hasse, H.: Über die Klassenzahl des Körpers mit einer Primzahlp≡1 mod 23. Aequationes Math.3, 165–169 (1969).

    Google Scholar 

  35. Hasse, H.: Über die Teilbarkeit durch 23 der Klassenzahl imaginär-quadratischer Zahlkörper mit genau zwei verschiedenen Diskriminantenprimteilern. J. Reine Angew. Math.241, 1–6 (1970).

    Google Scholar 

  36. Hasse, H. andJ. Liang: Über den Klassenkörper zum quadratischen Zahlkörper mit der Diskriminante−47 (Fortsetzung). Acta Arith.16, 89–97 (1969).

    Google Scholar 

  37. Hecke, E.: Vorlesungen über die Theorie der algebraischen Zahlen. Leipzig: Akademische Verlagsgesellschaft, 1923.

    Google Scholar 

  38. Hemer, O.: Notes on the Diophantine equationy 2−k=x3. Ark. Mat.3, 67–77 (1954).

    Google Scholar 

  39. Hemer, O.: On some Diophantine equations of the typey 2−f2=x3 Math. Scand.4, 95–107 (1956).

    Google Scholar 

  40. Hollkott, A.: Finite Konstruktion geordneter algebraischer Erweiterungen von geordneten Grundkörpern. Dissertation, Hamburg, 1941.

  41. Ince, E. L.: Cycles of reduced ideals in quadratic field. Brit. Assoc. Advancement Sci., Math. Tables IV (1934).

  42. Iwasawa, K. andC. C. Sims: Computation of invariants in the theory of cyclotomic fields. J. Math. Soc. Japan18, 86–96 (1966).

    Google Scholar 

  43. Kempfert, H.: On sign determination in real algebraic number fields. Numer. Math.11, 170–174 (1968).

    Google Scholar 

  44. Lehmer, D. H.: On imaginary quadratic fields whose class number is unity. Bull. Amer. Math. Soc.39, 360 (1933).

    Google Scholar 

  45. Lehmer, D. H., E. Lehmer, andD. Shanks: Integer sequences having prescribed quadratic character. Math. Comp.24, 433–453 (1970).

    Google Scholar 

  46. Leopoldt, H. W.: Über Einheitengruppe und Klassenzahl reeller abelscher Zahlkörper. Abh. Deutsch. Akad. Wiss. Berlin, Math.-Nat. Kl.2, 48 pp. (1954).

  47. Leopoldt, H. W.: Über ein Fundamentalproblem der Theorie der Einheiten algebraischer Zahlkörper. Bayer. Akad. Wiss., Math.-Nat. Kl. S.-B. (1956), 41–48 (1957).

  48. Leopoldt, H. W.: Über Klassenzahlprimteiler reeller abelscher Zahlkörper als Primteiler verallgemeinerterBernoullischer Zahlen. Abh. Math. Sem. Univ. Hamburg23, 36–47 (1959).

    Google Scholar 

  49. Leopoldt, H. W.: ÜberFermat-Quotienten von Kreiseinheiten und Klassenzahlformeln modp. Rend. Circ. Mat. Palermo (2)9, 39–50 (1960).

    Google Scholar 

  50. Leopoldt, H. W.: Klassenzahlen und Klassengruppen imaginär-quadratischer Zahlkörper mit Primzahldiskriminanteq=−1 mod 4. (Unpublished).

  51. Ljunggren, W.: On theDiophantine equationy 2−k=x3 Acta Arith.8, 451–463 (1962/63).

    Google Scholar 

  52. Ljunggren, W.: On theDiophantine equationAx 4−By2=C (C=1, 4). Math. Scand.21, 149–158 (1967).

    Google Scholar 

  53. Ljunggren, W.: A remark concerning the paper “On the Diophantine equationAx 4−By2=C (C=1, 4)”. Math. Scand.22, 282 (1968).

    Google Scholar 

  54. Macbeath, A. M.: Extensions of the rationals withGalois groupPGL (2,Z n). Bull. London Math. Soc.1, 332–338 (1969).

    Google Scholar 

  55. Matzat, B. H.: Zahlentheoretische Programme und einige Ergebnisse. (Unpublished).

  56. Rowlinson, E. andH. Schwerdtfeger: Polynomials with certain prescribed conditions on theGalois group. Canad. J. Math.21, 262–273 (1969).

    Google Scholar 

  57. Schaffstein, K.: Tafel der Klassenzahlen der reellen quadratischen Zahlkörper mit Primzahl Diskriminante unter 12,000 und zwischen 100,000–101,000 und 1,000,000–1,001,000. Math. Ann.98, 745–748 (1928).

    Google Scholar 

  58. Scholz, A. andO. Taussky: Die hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper: ihre rechnerische Bestimmung und ihr Einfluß auf den Klassenkörperturm. J. Reine Angew. Math.171, 19–41 (1934).

    Google Scholar 

  59. Selmer, E. S.: The Diophantine equationax 3+by3+cz3=0. Acta Math.85, 203–362 (1951).

    Google Scholar 

  60. Selmer, E. S.: TheDiophantine equationax 3+by3+cz3=0. Completion of the tables. Acta Math.92, 191–197 (1954).

    Google Scholar 

  61. Selmer, E. S.: TheDiophantine equation. A note onCassels method. Math. Scand.3, 68–74 (1955).

    Google Scholar 

  62. Selmer, E. S.: OnCassels' conditions for rational solubility of theDiophantine equation. Arch. Math. Naturvid.53, 115–137 (1956).

    Google Scholar 

  63. Selmer, E. S.: Tables for the purely cubic field. Avh. Norske Vid. Akad. Oslo I (1955),5, 38 pp. (1956).

  64. Selmer, E. S.: The rational solutions of theDiophantine equation for |D|≤100. Math. Scand.4, 281–286 (1956).

    Google Scholar 

  65. Shanks, D.: GeneralizedEuler and class numbers. Math. Comp.21, 689–694 (1967).

    Google Scholar 

  66. Shanks, D.: Corrigenda to: “GeneralizedEuler and class numbers”. Math. Comp.22, 699 (1968).

    Google Scholar 

  67. Shanks, D.: OnGauss' class number problems. Math. Comp.23, 151–163 (1969).

    Google Scholar 

  68. Shanks, D.: Class number, a theory of factorization, and genera. To appear.

  69. Slavutskii, I. Š.: On the class-number of the ideals of a real quadratic field. Izv. Vysš. Učebn. Zaved. Mat.4 (17), 173–177 (1960).

    Google Scholar 

  70. Slavutskii, I. Š.: On the ideal class number of a real quadratic field with prime discriminant. Leningrad Gos. Ped. Inst. Učen. Zap.218, 179–189 (1961).

    Google Scholar 

  71. Slavutskii, I. Š.: Upper bounds and numerical calculation of the number of ideal classes of real quadratic fields. Amer. Math. Soc. Transl. (2)82, 67–71 (1969).

    Google Scholar 

  72. Smith, D. L.: The calculation of simple continued-fraction expansions of real algebraic numbers. Master thesis, Columbus, Ohio, 1969.

  73. Stark, H. M.: An explanation of some exotic continued fractions found byJ. Brillhart. Proc. Atlas Sympos. No. 2, Oxford, England, 1969 (to appear).

  74. Stender, H.-J.: Explizite Bestimmung von Einheiten für einige Klassen algebraischer Zahlkörper. Dissertation, Köln, 1970.

  75. Stephens, N. M.: Conjectures concerning elliptic curves. Bull. Amer. Math. Soc.73, 160–163 (1967).

    Google Scholar 

  76. Stephens, N. M.: TheDiophantine equationX 3+Y3=D Z3 and the conjectures ofBirch andSwinnerton-Dyer. J. Reine Angew. Math.231, 121–162 (1968).

    Google Scholar 

  77. Stephens, N. M.: Completion of tables fory 2=x3+k (−100≤k<0) by a method ofLjunggren. Proc. Atlas Sympos. No. 2, Oxford, England, 1969 (to appear).

  78. Swinnerton-Dyer, H. P. F.: Table for class number and fundamental unit for cubic cyclic fields with prime conductorf<2,800, obtained on EDSAC (Cambridge) by a method inHasse [31], using a result of Heilbronn. (Unpublished).

  79. Swinnerton-Dyer, H. P. F.: An application of computing to class field theory. “Algebraic Number Theory”, Brighton: Proc. Instructional Conf. 1965, 280–291. Washington D. C.: Thompson, 1967.

    Google Scholar 

  80. Swinnerton-Dyer, H. P. F.: The conjectures ofBirch andSwinnerton-Dyer, and ofTate. 132–157. Proc. Conf. Local Fields Driebergen, 1966, Berlin-Heidelberg-New York: Springer. 1967.

    Google Scholar 

  81. Tate, J.: Algebraic cycles and poles of zeta functions. “Arithmetic Algebraic Geometry”, 93–110. Proc. Conf. Purdue Univ., 1963, New York. 1965.

  82. Taussky, O.: A remark concerningHilbert's theorem 94. J. Reine Angew. Math.239/240, 435–438 (1969).

    Google Scholar 

  83. Taussky, O.: Hilbert's theorem 94. Proc. Atlas Sympos. No. 2, Oxford, England, 1969 (to appear).

  84. Trinks, W.: Ein Beispiel eines Zahlkörpers mit derGalois-GruppePSL (3, F2) überQ. (Unpublished).

  85. Tuškina, T. A.: A numerical experiment on the calculation of theHasse invariant for certain curves. Amer. Math. Soc. Transl. (2)66, 204–205 (1968).

    Google Scholar 

  86. Yokoi, H.: On the fundamental unit of real quadratic fields with norm 1. J. Number Theory2, 106–114 (1970).

    Google Scholar 

  87. Zassenhaus, H.: Ein Algorithmus zur Berechnung einer Minimalbasis über gegebener Ordnung. ISNM7, 90–103 (1967).

    Google Scholar 

  88. Zassenhaus, H.: Continued fraction development of irrational real algebraic numbers. Columbus, Ohio, 1968 (unpublished).

  89. Zassenhaus, H.: A real root calculus. “Computational problems in abstract algebra”, 383–392. Proc. Conf. Oxford, 1967, Oxford-New York: Pergamon Press, 1970.

    Google Scholar 

  90. Zassenhaus, H.: The second round of the ORDMAX program. Columbus, Ohio, 1969 (unpublished).

  91. Zassenhaus, H.: On the units of orders. Columbus, Ohio, 1971 (unpublished).

  92. Zassenhaus, H., andH. Kempfert: The modified algorithm for the maximal order over a commutative order. (Unpublished).

  93. Zassenhaus, H., andJ. Liang: On a problem ofHasse. Math. Comp.23, 515–519 (1969).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute II of the University of Karlsruhe, D-75, Karlsruhe, Germany

    Horst G. Zimmer

Authors
  1. Horst G. Zimmer
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zimmer, H.G. Some computational aspects of, and the use of computers in, algebraic number theory. Computing 8, 363–381 (1971). https://doi.org/10.1007/BF02234117

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02234117

Keywords

Search

Navigation