Math 2.0

Number theory Definition

Witt group

Builds on Quadratic formCharacteristic of a ringAbelian groupWitt's theorem

Definition

Let FF be a field whose characteristic is not 22. A quadratic space over FF is a finite-dimensional vector space with a nondegenerate quadratic form. The hyperbolic plane HH is F2F^2 with the form q(x,y)=xyq(x, y) = xy. For quadratic spaces q1q_1 and q2q_2, write q1⊥q2q_1 \perp q_2 for their orthogonal sum, the form (v1,v2)↦q1(v1)+q2(v2)(v_1, v_2) \mapsto q_1(v_1) + q_2(v_2) on the direct sum. Write mHmH for the orthogonal sum of mm copies of HH. Two quadratic spaces q1,q2q_1, q_2 are Witt equivalent when q1⊥mHq_1 \perp mH and q2⊥nHq_2 \perp nH are isometric for some integers m,n≥0m, n \ge 0. The Witt group W(F)W(F) is the set of Witt equivalence classes of quadratic spaces over FF, with the operation [q1]+[q2]=[q1⊥q2][q_1] + [q_2] = [q_1 \perp q_2].

Remarks

This operation is well defined, and W(F)W(F) is an abelian group. The zero is the class of HH and of the zero space, and −[q]=[−q]-[q] = [-q], because q⊥(−q)q \perp (-q) is a sum of hyperbolic planes. Witt's theorem gives that each class contains exactly one anisotropic form up to isometry. Example: W(ℂ)≅ℤ/2ℤW(\C) \cong \Z/2\Z, given by the dimension modulo 22, because over ℂ\C the form x2+y2x^2 + y^2 is hyperbolic. Example: W(ℝ)≅ℤW(\R) \cong \Z, given by the signature. Non-example: the class of x2x^2 in W(ℝ)W(\R) is not zero, because x2x^2 is anisotropic and the zero class contains only the zero space as anisotropic form. With the tensor product of forms, W(F)W(F) is a commutative ring, the Witt ring.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Abelian group, concept, no evidence levelAbelian groupconceptCharacteristic of a ring, concept, no evidence levelCharacteristic of a r…conceptFinite-dimensional vector space, concept, no evidence levelFinite-dimensional ve…conceptLinear subspace, concept, no evidence levelLinear subspaceconceptNondegenerate bilinear form, concept, no evidence levelNondegenerate bilinea…conceptQuadratic form, concept, no evidence levelQuadratic formconceptSymmetric bilinear form, concept, no evidence levelSymmetric bilinear fo…conceptWitt's theorem, statement, no evidence levelWitt's theoremstatementWitt group, definition, no evidence levelWitt groupdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Abelian group concept
  2. Characteristic of a ring concept
  3. Finite-dimensional vector space concept
  4. Linear subspace concept
  5. Nondegenerate bilinear form concept
  6. Quadratic form concept
  7. Symmetric bilinear form concept
  8. Witt's theorem statement
  9. Witt group definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Witt_group
About this page
Tag7ABT36371X
URIm2://7ABT36371X
Kinddefinition
WritingW0 stub
MSC11E81, 19G12, 15A63
DefinesWitt group
UsesQuadratic form, Characteristic of a ring, Abelian group, Witt's theorem
Changed2 November 2026

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