Advertisement

The Ramanujan Journal

, Volume 48, Issue 1, pp 151–157 | Cite as

Universal quadratic forms over multiquadratic fields

  • Vítězslav KalaEmail author
  • Josef Svoboda
Article
  • 72 Downloads

Abstract

For all positive integers k and N, we prove that there are infinitely many totally real multiquadratic fields K of degree \(2^k\) over \(\mathbb {Q}\) such that each universal quadratic form over K has at least N variables.

Keywords

Universal quadratic form Multiquadratic number field Additively indecomposable integer 

Mathematics Subject Classification

11E12 11R20 

Notes

Acknowledgements

We wish to thank the anonymous referees for a careful reading of the manuscript and for several very helpful comments and corrections.

References

  1. 1.
    Bhargava, M.: On the Conway-Schneeberger fifteen Theorem. Contemp. Math. 272, 27–37 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bhargava, M., Hanke, J.: Universal quadratic forms and the 290-theorem. Invent. Math. (to appear)Google Scholar
  3. 3.
    Blomer, V., Kala, V.: Number fields without universal \(n\)-ary quadratic forms. Math. Proc. Camb. Philos. Soc. 159, 239–252 (2015)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chan, W.K., Icaza, M.I.: Positive definite almost regular ternary quadratic forms over totally real number fields. Bull. Lond. Math. Soc. 40, 1025–1037 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chan, W.K., Kim, M.-H., Raghavan, S.: Ternary universal integral quadratic forms. Jpn. J. Math. 22, 263–273 (1996)CrossRefzbMATHGoogle Scholar
  6. 6.
    Collinet, G.: Sums of squares in rings of integers with 2 inverted. Acta Arith. 173(4), 383–390 (2016)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Deutsch, J.I.: Universality of a non-classical integral quadratic form over \(\mathbb{Q}[\sqrt{5}]\). Acta Arith. 136, 229–242 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dress, A., Scharlau, R.: Indecomposable totally positive numbers in real quadratic orders. J. Number Theory 14, 292–306 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Earnest, A.G.: Universal and regular positive quadratic lattices over totally real number fields. In: Integral Quadratic Forms and Lattices (Seoul, 1998). Contemporary Mathematics, vol. 249, pp. 17–27. American Mathematical Society (1999)Google Scholar
  10. 10.
    Earnest, A.G., Khosravani, A.: Universal positive quaternary quadratic lattices over totally real number fields. Mathematika 44, 342–347 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jang, S.W., Kim, B.M.: A refinement of the Dress-Scharlau theorem. J. Number Theory 158, 234–243 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jarvis, F.: Algebraic Number Theory. Springer, New York (2014)zbMATHGoogle Scholar
  13. 13.
    Kala, V.: Universal quadratic forms and elements of small norm in real quadratic fields. Bull. Aust. Math. Soc. 94, 7–14 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kala, V.: Norms of indecomposable integers in real quadratic fields. J. Number Theory 166, 193–207 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kim, B.M.: Finiteness of real quadratic fields which admit positive integral diagonal septenary universal forms. Manuscr. Math. 99, 181–184 (1999)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kim, B.M.: Universal octonary diagonal forms over some real quadratic fields. Commentarii Math. Helv. 75, 410–414 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sasaki, H.: Quaternary universal forms over \(\mathbb{Q}[\sqrt{13}]\). Ramanujan J. 18, 73–80 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Schmal, B.: Diskriminanten, \({\mathbb{Z}}\)-Ganzheitsbasen und relative Ganzheitsbasen bei multiquadratischen Zahlkörpern. Arch. Math. 52, 245–257 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Siegel, C.L.: Sums of \(m\)-th powers of algebraic integers. Ann. Math. 46, 313–339 (1945)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Svoboda, J.: Universal quadratic forms over number fields. Bachelor’s Thesis, Charles University, Prague, iii+17 pp (2016)Google Scholar
  21. 21.
    Yatsyna, P.: A lower bound for the rank of a universal quadratic form with integer coefficients in a totally real field (preprint)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Algebra, Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic
  2. 2.Mathematisches InstitutUniversity of GöttingenGöttingenGermany

Personalised recommendations