PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-17

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 base-field embedding carries an algebraic element to a conjugate

Statement

Let $σ : K \to L$ be an $F$ -embedding and let $α \in K$ be algebraic over $F$ . Then $σ (α)$ is conjugate to $α$ over $F$ . In particular, an $F$ -endomorphism of a splitting field permutes the distinct roots of every base polynomial that splits there.

Facts & Assumptions

Given: An $F$ -embedding $σ : K \to L$ and an element $α \in K$ algebraic over $F$ .

[L1]

A field isomorphism transports polynomial evaluation and carries roots to roots after applying the induced coefficient map (A field isomorphism transports polynomials coefficientwise and carries roots, factorizations, and splitting to roots, factorizations, and splitting).

[L2]

An endomorphism of a splitting field fixing the base permutes the finite set of distinct roots of the defining polynomial (Every $F$ -endomorphism of a splitting field permutes the distinct roots and is an automorphism).

[L3]

Conjugate elements are the roots of the same minimal polynomial over the base (Conjugate algebraic elements over a field).

Proof

technique · direct

1.1L1

Regard $σ$ as an isomorphism $K \to σ (K) \subseteq L$ . Let $m_{α} \in F [x]$ be the minimal polynomial of $α$ . Since $σ$ fixes $F$ , [L1] gives $m_{α} (σ (α)) = σ (m_{α} (α)) = 0$ .

2.1step 1.1L3

Thus $σ (α)$ is a root of $m_{α}$ and is conjugate to $α$ by [L3].

3.1L2∎

When $K = L$ is a splitting field, [L2] strengthens this root preservation to a permutation of the distinct roots.

Depends on

Used by

Dependency tree · two levels

11 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

P. L. Clark, Field Theory, Chapters 3 to 5 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, Chapters 2, 3, and 5 (standard reference, not scraped)