Witt ring
To classify quadratic forms over a field, one wants a single algebraic gadget whose elements are forms and whose arithmetic mirrors how forms combine. The Witt ring is exactly that: package all anisotropic forms over a field into a ring, using orthogonal sum as addition and tensor product as multiplication, after agreeing to ignore the ‘trivial’ hyperbolic planes that carry no anisotropic content.
Construction: over a field K of characteristic not 2, consider nondegenerate quadratic forms up to isometry. Orthogonal sum gives a commutative monoid; Witt cancellation lets one form the Grothendieck group and then quotient by the ideal generated by the hyperbolic plane H. The result W(K) is a commutative ring: addition is orthogonal sum, multiplication is tensor product, and two forms are equal in W(K) iff their anisotropic cores are isometric. Each class has a unique anisotropic representative by Witt decomposition.
The Witt ring concentrates the arithmetic of a field. W(C) ≅ Z/2Z, W(R) ≅ Z via the signature, and W(F_q) is small and explicit for finite fields. The fundamental ideal I of even-dimensional forms gives a filtration whose graded pieces I^n/I^{n+1} are computed by Milnor K-theory mod 2 and by Galois cohomology — the Milnor conjecture, proved by Voevodsky — making W(K) a meeting point of quadratic forms, K-theory, and cohomology.
Over R, every anisotropic form is ±(positive definite), so W(R) ≅ Z with the class of an n-dimensional form of signature (p, q) mapping to p − q; the hyperbolic plane H of signature (1, 1) maps to 0, as it must.
The Witt ring of R is Z, recording the signature.
Subtlety: some authors call W(K) the Witt ring of quadratic forms and reserve a separate Witt-Grothendieck ring GW(K) for forms before killing hyperbolics; W(K) = GW(K) modulo the hyperbolic ideal.