CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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Over a field of characteristic not $2$ , every quadratic form has diagonal coordinates $q (x) = a_{1} x_{1}^{2} + \dots + a_{n} x_{n}^{2}$

Statement

Let $q$ be a quadratic form on an $n$ -dimensional vector space over a field of characteristic not $2$ . Some basis gives

$q (x_{1} e_{1} + \dots + x_{n} e_{n}) = a_{1} x_{1}^{2} + \dots + a_{n} x_{n}^{2}$

for scalars $a_{1}, \dots, a_{n}$ , which may include zeros.

Facts & Assumptions

Given: The quadratic form $q$ in the stated characteristic.

[L1]

Polarization gives a symmetric bilinear form $B_{q} = \frac{1}{2} b_{q}$ satisfying $q (v) = B_{q} (v, v)$ (If $char F \neq 2$ , quadratic forms and symmetric bilinear forms correspond by $q (v) = B (v, v)$ and $B (u, v) = \frac{1}{2} b_{q} (u, v)$ ).

[L2]

Every finite-dimensional symmetric bilinear form in characteristic not $2$ has an orthogonal basis (Every symmetric bilinear form on a finite-dimensional space over a field of characteristic not $2$ has an orthogonal basis).

Proof

technique · direct

1.1

Form $B_{q}$ by [L1] and choose an orthogonal basis $(e_{1}, \dots, e_{n})$ by [L2]. Set $a_{i} = B_{q} (e_{i}, e_{i})$ .

L1L2choose

2.1

For $v = \sum_{i} x_{i} e_{i}$ , bilinearity and orthogonality give $q (v) = B_{q} (v, v) = \sum_{i} a_{i} x_{i}^{2}$ ; every cross term vanishes.

step 1.1L1algebra

3.1

This is the required diagonal expression. Zero coefficients are allowed, so degenerate forms and the zero form are included; for $n = 0$ the sum is empty.

step 2.1∎

Depends on

Used by

q(x,y)=2x²+4xy+5y²=2(x+y)²+3y² has inertia (2,0,0) Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 36 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

K. Conrad, Bilinear Forms, §7 (standard reference, not scraped)

Over a field of characteristic not 22, every quadratic form has diagonal coordinates q(x)=a1x12+⋯+anxn2q(x)=a1​x12​+⋯+an​xn2​