LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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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.

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

Statement

Let $f \in R [x]$ and let $z \in C ∖ R$ . If $f (z) = 0$ , then $f (\overline{z}) = 0$ .

Facts & Assumptions

Given: A real polynomial $f \in R [x]$ 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

1.1

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

1.2

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

L1givenalgebra

2.1

Therefore $\overline{z}$ is also a root of $f$ . Since $z \notin R$ , one has $\overline{z} \neq z$ , so the two roots form a conjugate pair.

step 1.2algebra∎

Depends on

The only real-field automorphisms of $\mathbb C$ are the identity and complex conjugation

Used by

An irreducible polynomial in ℝ[x] has degree 1 or 2 Corollary

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

J. S. Milne, Fields and Galois Theory, v5.10, Chapter 5 (standard reference, not scraped)