TheoremStatement: 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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Alternating forms are skew-symmetric; the converse holds when $char F \neq 2$ , while in characteristic $2$ alternating forms are symmetric

Statement

Every alternating bilinear form is skew-symmetric. If $char F \neq 2$ , every skew-symmetric bilinear form is alternating. If $char F = 2$ , every alternating bilinear form is symmetric.

Facts & Assumptions

Given: A bilinear form $B : V \times V \to F$ .

[L1]

Alternating means $B (v, v) = 0$ for all $v$ , skew-symmetric means $B (u, v) = - B (v, u)$ , and symmetric means $B (u, v) = B (v, u)$ (Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms).

[L2]

The characteristic is the least positive natural multiple of $1_{F}$ that is zero, or $0$ if none exists (The characteristic of a ring: the least $n \geq 1$ with $n \cdot 1_{R} = 0$ when one exists, and $0$ otherwise); a field has $0_{F} \neq 1_{F}$ and every nonzero element is invertible (Field).

Proof

technique · direct

1.1

If $B$ is alternating, bilinearity gives $0 = B (u + v, u + v) = B (u, v) + B (v, u)$ , so $B (u, v) = - B (v, u)$ and $B$ is skew-symmetric.

L1algebra

1.2

If $B$ is skew-symmetric, then $B (v, v) = - B (v, v)$ , so $2 B (v, v) = 0$ . When $char F \neq 2$ , [L2] makes $2 = 1_{F} + 1_{F}$ nonzero and invertible, giving $B (v, v) = 0$ for every $v$ and hence alternation.

L1L2algebra

2.1

If $char F = 2$ and $B$ is alternating, [L2] gives $1_{F} + 1_{F} = 0_{F}$ and hence $- a = a$ for every $a \in F$ . Step 1.1 therefore gives $B (u, v) = - B (v, u) = B (v, u)$ , so $B$ is symmetric.

step 1.1L1L2algebra

3.1

These arguments prove each asserted implication without claiming that every symmetric form in characteristic $2$ is alternating.

step 1.1step 1.2step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 48 results over 15 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, §4 (standard reference, not scraped)

Alternating forms are skew-symmetric; the converse holds when char⁡F≠2charF=2, while in characteristic 22 alternating forms are symmetric