Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-27
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.

A nonreal root of a real polynomial comes with its complex conjugate

Statement

Let f∈R[x] and let z∈C∖R. If f(z)=0, then f(z‾)=0.

Facts & Assumptions

Given: A real polynomial f∈R[x] and a nonreal complex number z with f(z)=0.

[L1]

Complex conjugation is a field automorphism of C that fixes R pointwise (The only real-field automorphisms of C are the identity and complex conjugation).

Proof

technique · direct
1.1L1

By [L1], complex conjugation fixes every coefficient of f.

1.2L1givenalgebra

Apply conjugation to the equality f(z)=0. Because conjugation is a field automorphism and fixes the coefficients of f, this gives 0=f(z)‾=f(z‾).

2.1step 1.2algebra∎

Therefore z‾ is also a root of f. Since z∉R, one has z‾≠z, so the two roots form a conjugate pair.

Depends on

Used by

Dependency tree · two levels

5 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