Similarity of Quadratic Forms Over Global Fields in Characteristic 2

  • Zhengyao WuEmail author


Let K be a global function field of characteristic 2. For each non-trivial place v of K, let \( K_{v} \) be the completion of K at v. We show that if two non-degenerate quadratic forms are similar over every \( K_{v} \), then they are similar over K. This provides an analogue of the version for characteristic not 2 previously obtained by T.Ono.


Similarity Quadratic form Global field Characteristic 2 

Mathematics Subject Classification

Primary: 11E12 Secondary: 11E81 11E88 



The author is supported by National Natural Science Foundation of China (No.11701352) and Shantou University Scientific Research Foundation for Talents (No.130-760188). The author thanks Yong Hu for helpful discussions.


  1. 1.
    Baeza, R.: Comparing \(u\)-invariants of fields of characteristic \(2\). Bol. Soc. Brasil. Mat. 13(1), 105–114 (1982)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Berhuy, G., Frings, C., Tignol, J.-P.: Galois cohomology of the classical groups over imperfect fields. J. Pure Appl. Algebra 211(2), 307–341 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cortella, A.: Le principe de Hasse pour les similitudes de formes quadratiques et hermitiennes. In: Théorie des nombres, Année 1991/1992, Publ. Math. Fac. Sci. Besançon, 1–11. Univ. Franche-Comté, Besançon (1992)Google Scholar
  4. 4.
    Cortella, A.: Un contre-exemple au principe de Hasse pour les similitudes de formes bilinéaires. C. R. Acad. Sci. Paris Sér. I Math. 317(8), 707–710 (1993)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Elman, R., Karpenko, N., Merkurjev, A.: The algebraic and geometric theory of quadratic forms. Amer. Math. Soc. Coll. Publ. 56. Amer. Math. Soc., Providence, RI (2008)Google Scholar
  6. 6.
    Gille, P.: Invariants cohomologiques de Rost en caractéristique positive. K-Theory 21(1), 57–100 (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gille, P., Szamuely, T.: Central simple algebras and Galois cohomology. Camb. Stud. in Adv. Math., vol. 165. Cambridge University Press, Cambridge (2017)CrossRefGoogle Scholar
  8. 8.
    Kato, K.: Galois cohomology of complete discrete valuation fields. In: Algebraic \(K\)-theory, Part II (Oberwolfach, 1980), Lect. Notes in Math. 967: 215–238. Springer, Berlin (1982)Google Scholar
  9. 9.
    Knus, M.-A.: Quadratic and Hermitian forms over rings. Grun. der Math. Wiss., vol. 294. Springer, Berlin (1991)CrossRefGoogle Scholar
  10. 10.
    Knus, M.-A., Merkurjev, A., Rost, M., Tignol, J.-P.: The book of involutions. Amer. Math. Soc. Coll. Publ. 44. Am. Math. Soc., Providence, RI (1998)Google Scholar
  11. 11.
    Lewis, D., Unger, T., Van Geel, J.: The Hasse principle for similarity of Hermitian forms. J. Algebra 285(1), 196–212 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mammone, P., Tignol, J.-P., Wadsworth, A.: Fields of characteristic \(2\) with prescribed \(u\)-invariants. Math. Ann. 290(1), 109–128 (1991)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Milne, J.S.: Duality in the flat cohomology of a surface. Ann. Sci. École Norm. Sup. (4) 9(2), 171–201 (1976)MathSciNetCrossRefGoogle Scholar
  14. 14.
    O’Meara, O.: Introduction to quadratic forms. Clas. in Math. Springer, Berlin (2000)Google Scholar
  15. 15.
    Ono, T.: Arithmetic of orthogonal groups. J. Math. Soc. Jpn. 7, 79–91 (1955)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Pollak, B.: Orthogonal groups over global fields of characteristic \(2\). J. Algebra 15, 589–595 (1970)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Prasad, G., Rapinchuk, A.: Local-global principles for embedding of fields with involution into simple algebras with involution. Comment. Math. Helv. 85(3), 583–645 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsShantou UniversityShantouChina

Personalised recommendations