The theory cf quadratic forms and the intimately related theory of sym metrie bilinear forms have a lang and rich his tory, highlighted by the work of Legendre, Gauss, Minkowski, and Hasse. (Compare [Dickson] and [Bourbaki, 24, p. 185].) Our exposition will concentrate on the rela tively recent developments which begin with and are inspired by Witt's 1937 paper "Theorie der quadratischen Formen in beliebigen Körpern." We will be particularly interested in the work of A. Pfister and M. Knebusch. However, some older material will be described, particularly in Chapter II. The presentation is based on lectures by Milnor at the Institute for Ad vanced Study, and at Haverford College under the Phillips Lecture Pro gram, during the Fall of 1970, as weIl as Iectures at Princeton University il1 1966. We want to thank J. Cunningham, M. Knebusch, M. Kneser, A. Rosenberg, W. Scharlau and J.-P. Serre for helpful suggestions and corrections. Prerequisites. The reader should be familiar with the rudiments of algebra., incJuding for example the concept of tensor product for mo dules over a commutative ring. A few individual sections will require quite a bit more. The logical relationship between the various chapters can be roughly described by the diagram below. There are also five appendices, largely self-contained, which treat special topics. I. Arbitrary commutative rings I H. The ring of V. Miscellaneous IIl. Fields integers examples IV. Dedekind domains Contents Chapter r. Basie Coneepts . . . . . . . . I. Basic Concepts.- II. Symmetric Inner Product Spaces over Z.- III. Inner Product Spaces over a Field.- IV. Discrete Valuations and Dedekind Domains.- V. Some Examples.- Appendix 1. Quadratic Forms.- Appendix 2. Hermitian Forms.- Appendix 3. The Hasse-Minkowski Theorem.- Appendix 4. Gauss Sums, the Signature mod 8, and Quadratic Reciprocity.- Appendix 5. The Leech Lattice, and Other Lattices in Dimension 24.- Chronological Table.- References.- Special Notations. Professor John Milnor ist am Mathematics Department/Institute for Mathematical Sciences der State University of New York at Stony Brook, USA tätig.
