ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

How statement and proof provenance work

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.

$x^{3} - 2$ has discriminant $- 108$ and Galois group $S_{3}$ over $Q$

Example

$x^{3} - 2$ has Galois group $S_{3}$ over $Q$ . Its discriminant is $- 108$ .

Facts & Assumptions

Given: Eisenstein's irreducibility criterion at $2$ (Eisenstein criterion over the integers) and the explicit splitting-field generators in The full $S_{3}$ correspondence for the splitting field of $x^{3} - 2$ .

[L1]

A monic irreducible separable cubic over a field of characteristic not two has group $S_{3}$ when its discriminant is not a square (A monic irreducible separable cubic in characteristic not two has Galois group $A_{3}$ or $S_{3}$ according to its discriminant).

Verification

technique · direct

1.1

Eisenstein at $2$ proves that $x^{3} - 2$ is irreducible over $Q$ .

given

1.2

Its depressed-cubic discriminant is $- 27 (- 2)^{2} = - 108$ , which is nonzero and negative, hence is not a square in $Q$ .

algebra

2.1

By [L1], steps 1.1 and 1.2 give Galois group $S_{3}$ , agreeing with the explicit six automorphisms of the splitting-field lattice example.

step 1.1step 1.2L1given∎

Depends on

Used by

FALSE: the degree of a polynomial determines its Galois group False statement

Dependency tree · two levels

20 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

K. Conrad, The Galois Correspondence, Example 4.6 (standard reference, not scraped)
K. Conrad, Galois Groups of Cubics and Quartics, Section 2 (standard reference, not scraped)

x3−2x3−2 has discriminant −108−108 and Galois group S3S3​ over QQ