Pacific Journal of Mathematics

Class numbers of imaginary cyclic quartic fields and related quaternary systems.

Richard H. Hudson

Article information

Source
Pacific J. Math., Volume 115, Number 1 (1984), 129-142.

Dates
First available in Project Euclid: 8 December 2004

Permanent link to this document
https://projecteuclid.org/euclid.pjm/1102708415

Mathematical Reviews number (MathSciNet)
MR762205

Zentralblatt MATH identifier
0548.12002

Subjects
Primary: 11D09: Quadratic and bilinear equations
Secondary: 11E20: General ternary and quaternary quadratic forms; forms of more than two variables 11R16: Cubic and quartic extensions

Citation

Hudson, Richard H. Class numbers of imaginary cyclic quartic fields and related quaternary systems. Pacific J. Math. 115 (1984), no. 1, 129--142. https://projecteuclid.org/euclid.pjm/1102708415


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References

  • [1] Adrian Albert, The integers of normal quartic fields, Annals of Math., 31 (1930), 381-418.
  • [2] Bruce Berndt, Classical theorems on quadratic residues, Enseignement Math., 22 (1976), 261-304.
  • [3] Bruce C. Berndt and Ronald J. Evans, Sums of Gauss, Jacobi, and Jacobsthal, J. Number Theory, 11 (1979), 349-398.
  • [4] Duncan A. Buell and Richard H. Hudson, Solutions of certain quaternary quadratic systems, Pacific J. Math., 114 (1984), 23-45.
  • [5] A. Cauchy, Mem.Institut de France, 17 (1840),697; Oeuvres,(1),III,388, Comptes Rendus, Paris 10 (1840), 451.
  • [6] L. E. Dickson, Cyclotomy and trinomial congruences, Trans. Amer. Math. Soc, 37 (1935), 363-380.
  • [7] P. G. Lejeune Dirichlet, Recherches sur diverses applications de analyse infinitesimale a la theorie des nombres, J. Reine Angew Math., 281 (1840),134-155.
  • [8] Hugh Edgar and Brian Peterson, Some contributions to the theory of cyclic quartic extensions of the rationals, J. Number Theory, 12 (1980), 77-83.
  • [9] Helmut Hasse, Uber die Klassenzahl abelscher Zahlkorper Akademie-Verlag, Berlin, 1952.
  • [10] Richard H. Hudson, Kenneth S. Williams, and Duncan A. Buell, Extension of a Theorem of Cauchy andJacobi, submitted for publication.
  • [11] Richard H. Hudson and Kenneth S. Williams, A class number formula for certain quartic fields, Carleton Mathematical Series, No. 174, 1981, Carleton University, Ottawa, Canada.
  • [12] Emma Lehmer, On Euler's criterion, J. Austral. Math. Soc, 1 (1959),64-70.
  • [13] Joseph B. Muskat and Yun-Cheng Zee, On the uniqueness of solutions of certain Diophantine equations, Proc. Amer. Math. Soc, 49 (1975),13-19.
  • [14] W. Narkiewicz,Elementary and Analytic Theory of Algebraic Numbers, Polish Scien- tific Publishers, Warsaw, 1974.
  • [15] Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, John Wiley and Sons Inc., New York, 1972.
  • [16] Bennett Setzer, The determination of all imaginary, quartic, Abelian numberfields with class number 1, Math. Comp., 35 (1980),1383-1386.
  • [17] Albert Leon Whiteman, Theorems analogous to Jacobsthas theorem, Duke Math., 16 (1949),619-626.