Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Similarity of quadratic forms and isomorphism of their function fields


Author: Adrian R. Wadsworth
Journal: Trans. Amer. Math. Soc. 208 (1975), 352-358
MSC: Primary 10C05; Secondary 15A63
DOI: https://doi.org/10.1090/S0002-9947-1975-0376527-8
MathSciNet review: 0376527
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper considers the question: Given anisotropic quadratic forms $ Q$ and $ Q'$ over a field $ K$ (char $ K \ne 2$), if their function fields are isomorphic must $ Q$ and $ Q'$ be similar? It is proved that the answer is yes if $ Q$ is a Pfister form or the pure part of a Pfister form, or a $ 4$-dimensional form. The argument for Pfister forms and their pure parts does not generalize because these are the only anisotropic forms which attain maximal Witt index over their function fields. To handle the $ 4$-dimensional case the following theorem is proved: If $ Q$ and $ Q'$ are two $ 4$-dimensional forms over $ K$ with the same determinant $ d$, then $ Q$ and $ Q'$ are similar over $ K$ iff they are similar over $ K[\sqrt d ]$. The example of Pfister neighbors suggests that quadratic forms arguments are unlikely to settle the original question for other kinds of forms.


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

  • [1] M. Knebusch, Specialization of quadratic and symmetric bilinear forms, and a norm theorem, Acta Math. 24 (1973), 279-299. MR 0349582 (50:2075)
  • [2] T. Y. Lam, The algebraic theory of quadratic forms, Benjamin, Reading, Mass., 1973. MR 0396410 (53:277)
  • [3] F. Lorenz, Quadratische Formen über Körpern, Lecture Notes in Math., vol. 130, Springer-Verlag, Berlin and New York, 1970. MR 44 #189. MR 0282955 (44:189)
  • [4] O. T. O'Meara, Introduction to quadratic forms, Die Grundlehren der Math. Wissenschaften, Band 117, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 27 #2485.
  • [5] T. Ono, Arithmetic of orthogonal groups, J. Math. Soc. Japan 7 (1955), 79-91. MR 16, 1087. MR 0069823 (16:1087a)
  • [6] A. Pfister, Multiplicative quadratische Formen, Arch. Math. 16 (1965), 363-370. MR 32 #2408. MR 0184937 (32:2408)
  • [7] E. Witt, Über ein Gegenbeispiel zum Normensatz, Math. Z. 39 (1935), 462-467. MR 1545510

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 10C05, 15A63

Retrieve articles in all journals with MSC: 10C05, 15A63


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0376527-8
Keywords: Similar quadratic forms, Pfister form, function field
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society