  Bott periodicity plays an important role in topological K-theory. The purpose
of this paper is to extend the periodicity theorem in a discrete context, where
all classical groups are involved and not just the general linear group. The
present paper generalizes previous results of the author [K1] and [K2], where 2
was assumed to be invertible in the rings involved. For the proof, two
important ideas have to be mentioned : the first one is due to Ranicki [R] who
introduced a kind of "enlarged" orthogonal group ; the second one is a genuine
cup-product between quadratic forms due to Clauwens [C]. As an example of
results obtained, we prove that the higher Witt groups of a finite field of
characteristic 2 are all isomorphic to Z/2. They generalize in some sense the
Dickson and Arf invariants.
