Abstract
This is a continuation of our previous article (Lorenz and Roquette in Math. Semesterber. 57(1):73–102, 2010) in this journal, where we discussed the paper of Cahit Arf (J. Reine Angew. Math. 183:148–167, 1941). There he introduced what today is called the “Arf invariant” of a quadratic form over a field of characteristic 2.
After our article had appeared we obtained some new information about the present state of the theory. When we say “new” then this means that this was new to us, i.e., we had not been aware before of the literature on the subject. (See our list of references.) It seems worthwhile to us to report here about it, as far as it is relevant for the assessment of Arf’s paper in its historical perspective.
Similar content being viewed by others
References
Aravire, B., Jacob, B.: p-algebras over maximally complete fields. Proc. Symp. Pure Math. 58(2), 27–49 (1995)
Aravire, R., Jacob, B.: Versions of Springer’s theorem for quadratic forms in characteristic 2. Am. J. Math. 118(2), 253–261 (1996)
Arf, C.: Untersuchungen über quadratische Formen in Körpern der Charakteristik 2. I. J. Reine Angew. Math. 183, 148–167 (1941)
Baeza, R.: Quadratic Forms over Semilocal Rings. Lecture Notes in Mathematics, vol. 655. Springer, Berlin (1978). VI, 199 pp.
Baeza, R.: Comparing u-invariants of fields in characteristic 2. Bol. Soc. Bras. Mat. 13(1), 105–114 (1982)
Draxl, P.K.: Über gemeinsame quadratische Zerfällungskörper von Quaternionenalgebren. Nachr. Akad. Wiss. Gött. Math.-Phys. Kl., 3F 16, 251–259 (1975)
Elman, R., Karpenko, N., Merkurjev, A.: The Algebraic and Geometric Theory of Quadratic Forms. Colloquium Publications, vol. 56. American Mathematical Society, Providence (2008). 435 pp
Faivre, F.: Liaison des formes de Pfister et corps de fonctions de quadriques en caractéristique 2. Ph.D. thesis, Université Franche-Comté (2006), 197 pp
Lam, T.Y.: On the linkage of quaternion algebras. Bull. Belg. Math. Soc. Simon Stevin 9, 415–418 (2002)
Lorenz, F., Roquette, P.: On the Arf invariant in historical perspective. Math. Semesterber. 57(1), 73–102 (2010)
Roquette, P.: The Brauer-Hasse-Noether Theorem in Historical Perspective. Schriftenreihe der Heidelberger Akademie der Wissenschaften, vol. 15. Springer, Berlin (2005). I, 77 pp
Teichmüller, O.: p-Algebren. Dtsch. Math. 1, 362–388 (1936)
Witt, E.: Theorie der quadratischen Formen in beliebigen Körpern. J. Reine Angew. Math. 176, 31–44 (1937)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lorenz, F., Roquette, P. On the Arf invariant in historical perspective, part 2. Math Semesterber 58, 125–136 (2011). https://doi.org/10.1007/s00591-011-0085-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00591-011-0085-y