State and explain Witt’s Cancellation Theorem and Witt’s Extension Theorem for quadratic forms (over \(\mathbb{R}\) and general fields).
Solution
Assume throughout that the field \(k\) has characteristic different from \(2\) and that the quadratic spaces are finite-dimensional and nondegenerate.
Witt extension theorem (Witt’s lemma). Let \((V,q)\) be a nondegenerate quadratic space. If \(U,U'\subseteq V\) and \[ f:U\xrightarrow{\sim}U' \] is an isometry, then there exists an isometry \[ \widetilde f\in O(V,q) \] whose restriction to \(U\) is \(f\).
A standard proof proceeds by induction on \(\dim U\), using reflections in anisotropic vectors and hyperbolic pairs to extend the isometry one vector at a time. The subspace \(U\) may be degenerate; only \(V\) is assumed nondegenerate.
Witt cancellation theorem. If \[ U\perp W_1\cong U\perp W_2 \] for nondegenerate quadratic spaces, then \[ W_1\cong W_2. \] More generally, if \(U_1\cong U_2\) and \[ U_1\perp W_1\cong U_2\perp W_2, \] then \(W_1\cong W_2\).
To see cancellation from extension, identify the two copies of \(U\) inside an ambient isometric quadratic space. The isometry between those copies extends by Witt’s theorem to an ambient isometry. After composing with that extension, the original ambient isometry fixes \(U\); therefore it carries the orthogonal complement \(U^\perp=W_1\) onto the orthogonal complement \(U^\perp=W_2\).
Over \(\mathbb R\), Sylvester’s law of inertia gives the concrete classification: every nondegenerate quadratic form is isometric to \[ x_1^2+\cdots+x_p^2-y_1^2-\cdots-y_q^2, \] and the pair \((p,q)\) is a complete isometry invariant. Witt cancellation is then also immediate from additivity of signature under orthogonal direct sum.