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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'' of characteristic not two. All vector spaces will be assumed to be finite-dimensional. We say that 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.〔Milnor & Husemoller (1973) p. 14〕 Each class is represented by the core form of a Witt decomposition.〔Lorenz (2008) p. 30〕
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.〔Milnor & Husemoller (1973) p. 65〕 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;〔Lorenz (2008) p. 37〕〔Milnor & Husemoller (1973) p. 72〕 the torsion subgroup is the kernel of the functorial map from ''W''(''k'') to ''W''(''k''py), where ''k''py is the Pythagorean closure of ''k'';〔Lam (2005) p. 260〕 it is generated by the Pfister forms.〔Lam (2005) p. 395〕 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.〔Lam (2005) p. 395〕
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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