ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

$F_{8} = F_{2} [t] / (t^{3} + t + 1)$ and its power table

Example

Let $a$ be the residue class of $t$ in $F_{2} [t] / (t^{3} + t + 1)$ . Then

$a^{3} = a + 1, a^{4} = a^{2} + a, a^{5} = a^{2} + a + 1, a^{6} = a^{2} + 1, a^{7} = 1.$

Together with $0$ and the low powers $a^{0} = 1$ , $a^{1} = a$ , $a^{2}$ , these are all eight elements.

Facts & Assumptions

Given: The quotient $A = F_{2} [t] / (t^{3} + t + 1)$ and the class $a$ of $t$ .

[L1]

A polynomial quotient over a field is a field exactly when its modulus is irreducible (For a nonconstant $p$ in $F [x]$ , the ideal $(p)$ is maximal and $F [x] / (p)$ is a field exactly when $p$ is irreducible).

[L2]

An irreducible cubic simple extension has power basis $(1, a, a^{2})$ and degree $3$ (A simple algebraic extension is its minimal-polynomial quotient and has power basis $1, a, \dots, a^{n - 1}$ and degree $n$ ).

[L3]

The ring $Z / 2$ is a field (For every prime $p$ , the two operations on $Z / p$ make it a field).

Verification

technique · direct

1.1

A reducible cubic over a field has a linear factor. The polynomial $t^{3} + t + 1$ has value $1$ at both elements of $F_{2}$ , so it is irreducible; [L1] and [L3] make $A$ a field.

givenL1L3

2.1

By [L2], the eight coefficient triples in the basis $(1, a, a^{2})$ are the eight elements of $A$ , and $a^{3} = a + 1$ .

step 1.1L2algebra

3.1

Successive multiplication by $a$ and reduction by $a^{3} = a + 1$ gives the displayed powers. Together with $a^{0} = 1$ , $a^{1} = a$ and $a^{2}$ , the powers $a^{0}, \dots, a^{6}$ are the seven distinct nonzero basis combinations, and the next product returns $a^{7} = 1$ .

step 2.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 73 results over 14 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, Finite Fields, Section 1 (standard reference, not scraped)

F8=F2[t]/(t3+t+1)F8​=F2​[t]/(t3+t+1) and its power table