ExampleConstruction: Literature-sourcedVerification: Literature-sourcedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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.

$Φ_{7}$ factors over $F_{2}$ into the two monic irreducible cubics

Example

Over $F_{2}$ one has

$Φ_{7} (t) = t^{6} + t^{5} + t^{4} + t^{3} + t^{2} + t + 1 = (t^{3} + t + 1) (t^{3} + t^{2} + 1) .$

These are the two monic irreducible cubic factors, and over a splitting field the two factors correspond to the Frobenius orbits

${ζ, ζ^{2}, ζ^{4}} and {ζ^{3}, ζ^{6}, ζ^{5}}$

of a primitive seventh root of unity $ζ$ .

Facts & Assumptions

Given: The field $F_{2}$ and a primitive seventh root of unity $ζ$ in a splitting field of $Φ_{7}$ .

[L1]

If $\gcd (n, q) = 1$ , the reduction of $Φ_{n}$ in $F_{q} [t]$ is a product of distinct monic irreducibles, each of degree the order of $[q]$ modulo $n$ (For $\gcd (n, q) = 1$ the reduction of $Φ_{n}$ in $F_{q} [t]$ is a product of distinct monic irreducibles, each of degree the order of $[q]$ modulo $n$ ).

[L2]

For $\gcd (n, q) = 1$ , the image of $Gal (F_{q} (μ_{n}) / F_{q})$ in $(Z / n)^{\times}$ is generated by $[q]$ (For $\gcd (n, q) = 1$ the image of $Gal (F_{q} (μ_{n}) / F_{q})$ in $(Z / n)^{\times}$ is generated by $[q]$ ).

[L3]

The roots of a monic irreducible polynomial of degree $d$ over $F_{q}$ are one Frobenius orbit $α, α^{q}, \dots, α^{q^{d - 1}}$ (A monic irreducible of degree $d$ over $F_{q}$ has the $d$ distinct roots $α, α^{q}, \dots, α^{q^{d - 1}}$ ).

Verification

technique · direct

1.1

In $(Z / 7)^{\times}$ the class $[2]$ has order $3$ , since $2^{3} = 8 \equiv 1 (m o d 7)$ and $2 \equiv̸ 1$ , $2^{2} = 4 \equiv̸ 1 (m o d 7)$ . So [L1] and [L2] say every irreducible factor of $Φ_{7}$ over $F_{2}$ is a distinct cubic.

L1L2algebra

2.1

The product of the two monic cubics is $(t^{3} + t + 1) (t^{3} + t^{2} + 1) = t^{6} + t^{5} + t^{4} + t^{3} + t^{2} + t + 1 = Φ_{7} (t)$ in $F_{2} [t]$ . Since both factors are monic of degree $3$ , step 1.1 makes them the two irreducible factors of $Φ_{7}$ .

step 1.1algebra

3.1

Frobenius acts by $ζ \mapsto ζ^{2}$ , so the orbit of $ζ$ is ${ζ, ζ^{2}, ζ^{4}}$ because $ζ^{8} = ζ$ , and the orbit of $ζ^{3}$ is ${ζ^{3}, ζ^{6}, ζ^{5}}$ because $(ζ^{3})^{2} = ζ^{6}$ , $(ζ^{6})^{2} = ζ^{12} = ζ^{5}$ and $(ζ^{5})^{2} = ζ^{10} = ζ^{3}$ . These are the two size-three Frobenius orbits of primitive seventh roots, and [L3] identifies them as the respective root sets of the two irreducible cubic factors from step 2.1.

step 1.1step 2.1L3algebra∎

Remarks

This is the degree-three case of the theorem, not a coincidence of cubics. The orbit size and the factor degree are both the order of $[2]$ modulo $7$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

38 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, Cyclotomic Extensions (expository blurb), Example 5.6 (standard reference, not scraped)
K. Conrad, Roots and Irreducibles (expository blurb), Example 6.2 (standard reference, not scraped)

Φ7Φ7​ factors over F2F2​ into the two monic irreducible cubics