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  "body": "\n## Definition\n\nLet $F$ be a field whose [[characteristic|0F6ES3BRYP]] is not $2$.\nA *quadratic space* over $F$ is a finite-dimensional vector space with a nondegenerate [[quadratic form|0RBQS1JJCR]].\nThe *hyperbolic plane* $H$ is $F^2$ with the form $q(x, y) = xy$.\nFor quadratic spaces $q_1$ and $q_2$, write $q_1 \\perp q_2$ for their orthogonal sum, the form $(v_1, v_2) \\mapsto q_1(v_1) + q_2(v_2)$ on the direct sum.\nWrite $mH$ for the orthogonal sum of $m$ copies of $H$.\nTwo quadratic spaces $q_1, q_2$ are *Witt equivalent* when $q_1 \\perp mH$ and $q_2 \\perp nH$ are isometric for some integers $m, n \\ge 0$.\nThe *Witt group* $W(F)$ is the set of Witt equivalence classes of quadratic spaces over $F$, with the operation $[q_1] + [q_2] = [q_1 \\perp q_2]$.\n\n## Remarks\n\nThis operation is well defined, and $W(F)$ is an [[abelian group|082GHNPJ53]].\nThe zero is the class of $H$ and of the zero space, and $-[q] = [-q]$, because $q \\perp (-q)$ is a sum of hyperbolic planes.\n[[Witt's theorem|CCB8VNQCV0]] gives that each class contains exactly one anisotropic form up to isometry.\nExample: $W(\\C) \\cong \\Z/2\\Z$, given by the dimension modulo $2$, because over $\\C$ the form $x^2 + y^2$ is hyperbolic.\nExample: $W(\\R) \\cong \\Z$, given by the signature.\nNon-example: the class of $x^2$ in $W(\\R)$ is not zero, because $x^2$ is anisotropic and the zero class contains only the zero space as anisotropic form.\nWith the tensor product of forms, $W(F)$ is a commutative ring, the Witt ring.\n",
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