Algebraic K-Theory and Quadratic Forms

The first section of this paper defines and studies a graded ring K . F associated to any field F. By definition, K~F is the target group of the universal n-linear function from F ~ x ... • F ~ to an additive group, satisfying the condition that al • " ' x a, should map to zero whenever a i -q-a i + ~ = 1. Here F ~ denotes the multiplicative group F 0 . Section 2 constructs a homomorphism ~: K,F---, K~__I_~ associated with a discrete valuation on F with residue class field F. These homomorphisms ~ are used to compute the ring K, F(t) of a rational function field, using a technique due to John Tate. Section 3 relates K . F to the theory of quadratic modules by defining certain " Stiefel-Whitney invariants" of a quadratic module over a field F of characteristic . 2 . The definition is closely related to Delzant [-5]. Let W be the Witt ring of anisotropic quadratic modules over F, and let I c W be the maximal ideal, consisting of modules of even rank. Section 4 studies the conjecture that the associated graded ring