Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Sums of three integer squares in complex quadratic fields

Authors: Dennis R. Estes and J. S. Hsia
Journal: Proc. Amer. Math. Soc. 89 (1983), 211-214
MSC: Primary 11E12; Secondary 11R11
MathSciNet review: 712624
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We classify all complex quadratic number fields that have all their algebraic integers expressible as a sum of three integer squares. These fields are $ F = {\mathbf{Q}}(\sqrt { - D} )$, $ D$ a positive square-free integer congruent to $ 3(\mod 8)$ and such that $ D$ does not admit a positive proper factorization $ D \equiv {d_1}{d_2}$ that satisfies simultaneously: $ {d_1} \equiv 5,7(\mod 8)$ and $ ({d_2}/{d_1}) = 1$.

References [Enhancements On Off] (What's this?)

  • [1] Dennis R. Estes and J. S. Hsia, Exceptional integers of some ternary quadratic forms, Adv. in Math. 45 (1982), no. 3, 310–318. MR 673805,
  • [2] H. Hasse, Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkorper, J. Reine Angew. Math. 153 (1924), 113-130.
  • [3] J. S. Hsia, Representations by integral quadratic forms over algebraic number fields, Conference on Quadratic Forms—1976 (Proc. Conf., Queen’s Univ., Kingston, Ont., 1976) Queen’s Univ., Kingston, Ont., 1977, pp. 528–537. Queen’s Papers in Pure and Appl. Math., No. 46. MR 0498377
  • [4] J. S. Hsia, On the classification of unimodular quadratic forms, J. Number Theory 12 (1980), no. 3, 327–333. MR 586460,
  • [5] O. T. O'Meara, Introduction to quadratic forms, Grundlehren der Math. Wiss., Springer-Verlag, New York, 1963.
  • [6] Philippe Revoy, Sur les sommes de carrés dans un anneau, Ann. Sci. Univ. Besançon Math. (3) 11 (1979), 3–8 (French, with English summary). MR 651967

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11E12, 11R11

Retrieve articles in all journals with MSC: 11E12, 11R11

Additional Information

Keywords: Exceptional integer, genus, $ \chi $-invariant, Artin symbol
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society