Two spaces equipped with symmetric bilinear forms are equivalent if one can be obtained from the other by adding a metabolic quadratic space, that is, zero or more copies of a hyperbolic plane, the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector.
[4] The elements of finite order in the Witt group have order a power of 2;[5][6] the torsion subgroup is the kernel of the functorial map from W(k) to W(kpy), where kpy is the Pythagorean closure of k;[7] it is generated by the Pfister forms
[8] If k is not formally real, then the Witt group is torsion, with exponent a power of 2.
[9] The height of the field k is the exponent of the torsion in the Witt group, if this is finite, or ∞ otherwise.
[9] It is a Noetherian ring if and only if there are finitely many square classes; that is, if the squares in k form a subgroup of finite index in the multiplicative group of k.[11] If k is not formally real, the fundamental ideal is the only prime ideal of W[12] and consists precisely of the nilpotent elements;[9] W is a local ring and has Krull dimension 0.
[13] If k is real, then the nilpotent elements are precisely those of finite additive order, and these in turn are the forms all of whose signatures are 0;[14] W has Krull dimension 1.
[5][15] If k is an ordered field with positive cone P then Sylvester's law of inertia holds for quadratic forms over k and the signature defines a ring homomorphism from W(k) to Z, with kernel a prime ideal KP.
[16] The n-th power of the fundamental ideal is additively generated by the n-fold Pfister forms.
[17] Certain invariants of a quadratic form can be regarded as functions on Witt classes.
The Hasse invariant of a quadratic form is again, a well-defined function on Witt classes with values in the Brauer group of the field of definition.
Addition and multiplication are defined by: Then there is a surjective ring homomorphism from W(K) to this obtained by mapping a class to discriminant and rank mod 2.
[23] The elements of Q may be regarded as classifying graded quadratic extensions of K.[24] The triple of discriminant, rank mod 2 and Hasse invariant defines a map from W(K) to the Brauer–Wall group BW(K).
For quadratic forms over K, there is a Hasse invariant ±1 for every finite place corresponding to the Hilbert symbols.
[27] The symbol ring is defined 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 is +1.
[28] Addition and multiplication is defined 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.
[30] The Hasse–Minkowski theorem implies that there is an injection[31] One can make this concrete and compute the image by using the "second residue homomorphism" W(Qp) → W(Fp).
Composed with the map W(Q) → W(Qp), one obtains a group homomorphism ∂p: W(Q) → W(Fp) (for p = 2, ∂2 is defined to be the 2-adic valuation of the discriminant, taken mod 2).
One will then have a split exact sequence[32] which can be written as an isomorphism where the first component is the signature.
form a descending filtration and one may consider the associated graded ring, that is the direct sum of quotients
Milnor showed also that this homomorphism sends elements divisible by 2 to 0 and that it is surjective.
In the same paper, he made a conjecture that this homomorphism is an isomorphism for all fields k (of characteristic different from 2).
The conjecture was proved by Dmitry Orlov, Alexander Vishik, and Vladimir Voevodsky[35] in 1996 (published in 2007) for the case
, leading to increased understanding of the structure of quadratic forms over arbitrary fields.
The Grothendieck-Witt ring GW is a related construction generated by isometry classes of nonsingular quadratic spaces with addition given by orthogonal sum and multiplication given by tensor product.
There is a natural homomorphism GW → Z given by dimension: a field is quadratically closed if and only if this is an isomorphism.
[18] The hyperbolic spaces generate an ideal in GW and the Witt ring W is the quotient.
[36] The exterior power gives the Grothendieck-Witt ring the additional structure of a λ-ring.
[42] Some variations and extensions of this condition, such as "tame degree l Hilbert symbol equivalence", have also been studied.
The quadratic L-groups are 4-periodic, with L0(R) being the Witt group of (1)-quadratic forms (symmetric), and L2(R) being the Witt group of (−1)-quadratic forms (skew-symmetric); symmetric L-groups are not 4-periodic for all rings, hence they provide a less exact generalization.