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.

$x^{3} - 3 x + 1$ has discriminant $81$ and Galois group $A_{3}$ over $Q$

Example

$x^{3} - 3 x + 1$ has Galois group $A_{3}$ over $Q$ . Its discriminant is $81$ , and its splitting field is a cyclic cubic extension of $Q$ .

Facts & Assumptions

Given: The rational-root theorem (Rational root theorem) and the discriminant convention of The discriminant of a monic polynomial as the coefficient expression of $Δ_{n}^{2}$ .

[L1]

A monic irreducible separable cubic over a field of characteristic not two has group $A_{3}$ when its discriminant is 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

The only rational-root candidates are $1$ and $- 1$ , and the polynomial takes the values $- 1$ and $3$ there. It has no rational root, so the cubic is irreducible.

given

1.2

For a depressed cubic $x^{3} + p x + q$ , the discriminant is $- 4 p^{3} - 27 q^{2}$ ; here it is $- 4 (- 3)^{3} - 27 = 81 = 9^{2}$ , which is nonzero.

algebra

2.1

Steps 1.1 and 1.2 give an irreducible separable cubic with square discriminant, so [L1] gives Galois group $A_{3}$ . Its order is three, equal to the splitting-field degree.

step 1.1step 1.2L1∎

Depends on

Used by

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

Dependency tree · two levels

18 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, v5.10, Example 4.7 (standard reference, not scraped)
K. Conrad, Galois Groups of Cubics and Quartics, Example 2.2 (standard reference, not scraped)

x3−3x+1x3−3x+1 has discriminant 8181 and Galois group A3A3​ over QQ