Skip to main content

Hermitian forms over division algebras over real function fields


We develop here the theory of (skew-) hermitian forms over division algebras over the real function field and its completions. In particular, a local and local-global classification for forms of all types are given and some Hasse principles are proved.

This is a preview of subscription content, access via your institution.


  • [B] Bartels, H.-J.:Invariaten hermiteschen Formen über Schiefkörpern, Math. Annalen 215, 269–288 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  • [D] Draxl, P.:Skew fields, London Math. Soc. Lec. Notes, Cambridge University Press, 1983

  • [E] Elman, R.:Rund forms over real algebraic function fields in one variable, Proc. Amer. Math. Soc. 41, 431–436 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  • [E-L] Elman, R. and Lam, T. Y.:Classification theorems for quadratic forms over fields, Comm. Math. Helv. 49, 373–381 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  • [G] Greenberg, M.:Lectures on forms in many variables, Benjamin, 1969

  • [H-J] Hsia, J. H. and Johnson, R. P.:Round and Pfister forms over R(t), Pacific J. Math. 49, 101–108 (1973)

    MATH  MathSciNet  Google Scholar 

  • [J] Jacobson, N.:Simple Lie algebras over a field of characteristic zero, Duke Math. J. 4, 534–551 (1938)

    Article  MATH  MathSciNet  Google Scholar 

  • [Ja] Janchevski, V. I.:Simple algebras with involution and unitary groups, Mat. Sb. 22, 372–385 (1974)

    Article  Google Scholar 

  • [K] Kneser, M.:Lecture on Galois cohomology of classical groups, Tata Ins. Fund. Research, Bombay, 1969

    Google Scholar 

  • [Kn] Knight, J.:Quadratic forms over R(t), Proc. Cambridge Philos. Soc. 62, 197–205 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  • [L] Lam, T. Y.:The algebraic theory of quadratic forms, Benjamin Reading, Mass., 1973

    MATH  Google Scholar 

  • [Le] Lewis, D. W.:Quaternionic Skew-Hermitian Forms over a Number Field, J. Algebra 74, 232–240 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  • [M-H] Milnor, J. W. and Husemoler, D.:Symmetric billinear forms, Berlin-Heidelberg-New York, Springer-Verlag, 1973

    Google Scholar 

  • [O] O'Meara, O. T.:Introduction to quadratic forms, Berlin-Heidelberg-New York, Springer, 1962

    Google Scholar 

  • [P] Pierce, R.:Associative algebras, GTM. no 88, Springer-Verlag 1982

  • [P1] Pollak, B.:Orthogonal groups over R((t)), Amer. J. Math. XC, 214–230 (1968)

    Article  MathSciNet  Google Scholar 

  • [P2] Pollak, B.: The equation tat=b in a quaternionalgebra, Duke Math. J. 27, 261–271 (1961)

    Article  MathSciNet  Google Scholar 

  • [Sch1] Scharlau, W.:Klassifikation hermiteschen Formen über lokalen Körpern, Math. Annalen 186, 201–208 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  • [Sch2] Scharlau, W.:Quadratic and Hermitian forms, Berlin-Heidelberg-New York, Springer-Verlag, 1985

    MATH  Google Scholar 

  • [Se1] Serre, J.-P.:Local fields, GTM n. 67, Springer-Verlag, 1979

  • [Se2] Serre, J.-P.:Cohomologie Galoisienne, LNM No. 5, Springer-Verlag 1965

  • [Ta] Tasaka, T.:On the quasi-split simple algebraic groups defined over an algebraic number field, J. Fac. Sci. Univ. Tokyo, Sec. IA, 20, 147–168 (1968)

    Google Scholar 

  • [Ti] Tits, J.:Classification of algebraic semisimple groups, in: Proc. Sym. Pure Mathematics v.IX, 33–62 (1966)

  • [Ts] Tsukamoto, T.:On the local theory of quaternionic antihermitian forms, J. Math. Soc. Jap. 13, 387–400 (1961)

    Article  MathSciNet  Google Scholar 

  • [W1] Witt, E.:Zerlegung reeller algebraischer Funktionen in Quadrate. Schiefkörper über reellem Funktionenkörper, J. Crelle 171, 4–11 (1934)

    MATH  Google Scholar 

  • [W2] Witt, E.:Theorie der quadratischen Formen in beliebigen Körpern, J. Crelle 176, 31–44 (1937)

    Google Scholar 

Download references

Author information

Authors and Affiliations


Rights and permissions

Reprints and Permissions

About this article

Cite this article

Thang, N.Q. Hermitian forms over division algebras over real function fields. Manuscripta Math 78, 9–35 (1993).

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI:


  • Quadratic Form
  • Division Algebra
  • Twisted Space
  • Hermitian Form
  • Global Field