Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: Literature-sourcedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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.

The polynomial x2+1x^2+1 is irreducible over R\mathbb R

Example

The polynomial x2+1x^2+1 is irreducible in R[x]\mathbb R[x].

Facts & Assumptions

Given: The polynomial f=x2+1∈R[x]f=x^2+1\in\mathbb R[x].

[L1]

A quadratic over a field is irreducible exactly when it has no root in that field (A polynomial of degree two or three over a field is irreducible exactly when it has no root in the field).

[L2]

The real numbers form an ordered field, so every square is nonnegative and 1>01>0 (The reals form a totally ordered field).

Verification

technique · direct
1.1

For every real aa, [L2] gives a2≥0a^2\ge0 and hence a2+1>0a^2+1>0.

givenL2algebra
2.1

Thus ff has no real root, and [L1] makes it irreducible in R[x]\mathbb R[x].

step 1.1L1∎

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: 30 results over 11 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