Grothendieck–Witt group

Remarks

Here $\langle a_1, \ldots, a_n \rangle$ denotes the diagonal form $a_1 x_1^2 + \cdots + a_n x_n^2$, with all $a_i \in F^\times$. By Witt cancellation, a consequence of Witt's theorem, the monoid $M(F)$ is cancellative, so $M(F) \to GW(F)$ is injective. The dimension gives a ring homomorphism $\dim \colon GW(F) \to \Z$. Let $H = \langle 1, -1 \rangle$ be the hyperbolic plane; the multiples of $[H]$ form an ideal, and $GW(F)/\Z[H]$ is the Witt ring $W(F)$. Example: $GW(\R) \cong \Z \oplus \Z$, via the numbers of positive and of negative entries in a diagonalization, by Sylvester's law of inertia. Non-example: in $W(\R)$ the classes of the zero space and of $H$ agree, but in $GW(\R)$ they differ, since $\dim H = 2$.