In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.
Definition Fix a field k of characteristic not equal to 2. All vector spaces will be assumed to be finite-dimensional. 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. Each class is represented by the core form of a Witt decomposition. The Witt group of k is the abelian group W(k) of equivalence classes of non-degenerate symmetric bilinear forms, with the group operation corresponding to the orthogonal direct sum of forms. It is additively generated by the classes of one-dimensional forms. Although classes may contain spaces of different dimension, the parity of the dimension is constant across a class and so rk: W(k) → Z/2Z is a homomorphism. The elements of finite order in the Witt group have order a power of 2; the torsion subgroup is the kernel of the functorial map from W(k) to W(kpy), where kpy is the Pythagorean closure of k; it is generated by the Pfister forms ⟨ ⟨ w ⟩ ⟩ = ⟨ 1 , − w ⟩ {\displaystyle \langle \!\langle w\rangle \!\rangle =\langle 1,-w\rangle } with w {\displaystyle w} a non-zero sum of squares. If k is not formally real, then the Witt group is torsion, with exponent a power of 2. The height of the field k is the exponent of the torsion in the Witt group, if this is finite, or ∞ otherwise.
Ring structure The Witt group of k can be given a commutative ring structure, by using the tensor product of quadratic forms to define the ring product. This is sometimes called the Witt ring W(k), though the term "Witt ring" is often also used for a completely different ring of Witt vectors. To discuss the structure of this ring one assumes that k is of characteristic not equal to 2, so that one may identify symmetric bilinear forms and quadratic forms. The kernel of the rank mod 2 homomorphism is a prime ideal, I, of the Witt ring termed the fundamental ideal. The ring homomorphisms from W(k) to Z correspond to the field orderings of k, by taking signature with respective to the ordering. The Witt ring is a Jacobson ring. 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. If k is not formally real, the fundamental ideal is the only prime ideal of W and consists precisely of the nilpotent elements; W is a local ring and has Krull dimension 0. 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; W has Krull dimension 1. If k is a real Pythagorean field then the zero-divisors of W are the elements for which some signature is 0; otherwise, the zero-divisors are exactly the fundamental ideal. 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. These prime ideals are in bijection with the orderings Xk of k and constitute the minimal prime ideal spectrum MinSpec W(k) of W(k). The bijection is a homeomorphism between MinSpec W(k) with the Zariski topology and the set of orderings Xk with the Harrison topology. The n-th power of the fundamental ideal is additively generated by the n-fold Pfister forms.
Examples The Witt ring of C, and indeed any algebraically closed field or quadratically closed field, is Z/2Z. The Witt ring of R is Z. The Witt ring of a finite field Fq with q odd is Z/4Z if q ≡ 3 mod 4 and isomorphic to the group ring (Z/2Z)[F*/F*2] if q ≡ 1 mod 4. The Witt ring of a local field with maximal ideal of norm congruent to 1 modulo 4 is isomorphic to the group ring (Z/2Z)[V] where V is the Klein 4-group. The Witt ring of a local field with maximal ideal of norm congruent to 3 modulo 4 is (Z/4Z)[C2] where C2 is a cyclic group of order 2. The Witt ring of Q2 is of order 32 and is given by
Z 8 [ s , t ] / ⟨ 2 s , 2 t , s 2 , t 2 , s t − 4 ⟩ {\displaystyle \mathbf {Z} _{8}[s,t]/\langle 2s,2t,s^{2},t^{2},st-4\rangle } .
Invariants Certain invariants of a quadratic form can be regarded as functions on Witt classes. Dimension mod 2 is a function on classes: the discriminant is also well-defined. 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.
Rank and discriminant A ring is defined over K, Q(K), as a set of pairs (d, e) with d in K*/K* 2 and e in Z/2Z. Addition and multiplication are defined by:
( d 1 , e 1 ) + ( d 2 , e 2 ) = ( ( − 1 ) e 1 e 2 d 1 d 2 , e 1 + e 2 ) {\displaystyle (d_{1},e_{1})+(d_{2},e_{2})=((-1)^{e_{1}e_{2}}d_{1}d_{2},e_{1}+e_{2})}
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