ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The discriminant of $x^{2} + b x + c$ and its double-root criterion

Example

For $f (t) = t^{2} + b t + c$ over a field,

$Disc (f) = b^{2} - 4 c .$

This vanishes exactly when $f$ has a repeated root, in every characteristic.

Facts & Assumptions

Given: A field $F$ , a monic quadratic $f (t) = t^{2} + b t + c$ , and roots $α, β$ in a splitting field.

[L1]

The discriminant of a split monic quadratic is $(α - β)^{2}$ and vanishes exactly for a repeated root (The discriminant is $\prod_{i < j} (α_{i} - α_{j})^{2}$ and vanishes exactly when a monic polynomial has a repeated root).

[L2]

Vieta's formulas give $α + β = - b$ and $α β = c$ (Vieta's formulas identify the coefficients of a split monic polynomial with elementary symmetric functions of its roots).

Verification

technique · direct

1.1givenalgebra

Expand $(α - β)^{2} = (α + β)^{2} - 4 α β$ .

2.1step 1.1L1L2algebra

Substitute [L2] into step 1.1 to get $(α - β)^{2} = b^{2} - 4 c$ , and apply [L1] to identify this with $Disc (f)$ .

3.1L1algebra∎

By [L1], this element vanishes exactly when $α = β$ . In characteristic two the formula becomes $b^{2}$ , while $f^{'} = b$ , so the same repeated-root criterion remains valid; completing a square is a separate issue.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Fields and Galois Theory, discriminant discussion (standard reference, not scraped)

The discriminant of x2+bx+cx2+bx+c and its double-root criterion