Witt group

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". All vector spaces will be assumed to be finite-dimensional. We say that two symmetric bilinear forms are equivalent if one can be obtained from the other by adding zero or more copies of a hyperbolic plane (the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector). The Witt group of k is the abelian group of equivalence classes of non-degenerate symmetric bilinear forms, with the group operation corresponding to the orthogonal direct sum of forms.

The Witt group of "k" can be given a commutative ring structure, by using the tensor product of two bilinear forms to define the ring product. This is sometimes called the Witt ring of "k", though the term "Witt ring" is often also used for a completely different ring of Witt vectors.

Witt equivalence

Two fields are said to be Witt equivalent if their Witt rings are isomorphic. Two number fields "K" and "L" are Witt equivalent if and only if there is a bijection "T" between the places of "K" and the places of "L" and a group isomorphism "t" between their square-class groups, preserving degree 2 Hilbert symbols. In this case the pair ("T","t") is called a reciprocity equivalence or a degree 2 Hilbert symbol equivalence. Some variations and extensions of this condition, such as "tame degree "l" Hilbert symbol equivalence", have also been studied; see the references for details.

References

*S. Lang, "Algebra", ISBN 0-387-95385-X
* [http://eom.springer.de/W/w098080.htm Witt rings] in the Springer encyclopedia of mathematics
*Czogała, A. "Higher degree tame Hilbert-symbol equivalence of number fields." Abh. Math. Sem. Univ. Hamburg 69 (1999), 175-185.