Witt Group - Witt Ring of A Number Field

Witt Ring of A Number Field

Let K be a number field. For quadratic forms over K, there is a Hasse invariant ±1 for every finite place corresponding to the Hilbert symbols. The invariants of a form over a number field are precisely the dimension, discriminant, all local Hasse invariants and the signatures coming from real embeddings.

We define the symbol ring over K, Sym(K), as a set of triples (d,e,f) with d in K*/K*2, e in Z/2 and f a sequence of elements ±1 indexed by the places of K, subject to the condition that all but finitely many terms of f are +1, that the value on acomplex places is +1 and that the product of all the terms in f in +1. Let be the sequence of Hilbert symbols: it satisfies the conditions on f just stated.

We define addition and multiplication as follows:

Then there is a surjective ring homomorphism from W(K) to Sym(K) obtained by mapping a class to discriminant, rank mod 2, and the sequence of Hasse invariants. The kernel is I3.

The symbol ring is a realisation of the Brauer-Wall group.