Corollary 2

Tag 0DKW, permanent. Cite it as [Bourbaki, Tag 0DKW].

Algebra, chapter IX, § 4. Sous-espaces totalement isotropes. Théorème de Witt

Let Q be a non-degenerate quadratic form on a vector space E of finite dimension n over an algebraically closed field A; there then exists a basis (e_i)_{1 \leq i \leq n} of E such that

(2)

Q(\sum_{i=1}^{n} x_i e_i) = \sum_{i=1}^{\nu} x_i x_{i+\nu} \qquad si\ n = 2\nu,

(3)

Q(\sum_{i=1}^{n} x_i e_i) = \sum_{i=1}^{\nu} x_i x_{i+\nu} + x_{2\nu+1}^2 \qquad si\ n = 2\nu + 1.

Indeed, let F_1 and F_2 be two maximal totally singular subspaces such that F_1 ∩ F_2 = {0} (Cor. 1) and let q be their dimension. We then have G = {0} with the notation of Prop. 3. Taking a basis (e_i)_{1 \leq i \leq q} of F_1 and a basis (e_i)_{q+1 \leq i \leq 2q} of F_2 such that Φ(e_i, e_{j+q}) = δ_{ij} for i, j = 1, ..., q (Prop. 2 a)), we see that it suffices to show that dim H ≤ 1. Now, if x ∈ H, y ∈ H and if x ≠ 0, the equation Q(y - ax) = Q(y) - aΦ(x, y) + a^2 Q(x) = 0 has at least one solution a_0 since Q(x) ≠ 0, and we have y = a_0 x since every singular vector of H is zero.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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