TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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$F$ -embeddings of $F (α)$ into an algebraically closed field correspond to the distinct roots of $m_{α}$

Statement

Let $α$ be algebraic over $F$ , and let $Ω$ be an algebraically closed field containing $F$ . Sending an $F$ -embedding $σ : F (α) \to Ω$ to $σ (α)$ is a bijection from the set of such embeddings to the set of distinct roots in $Ω$ of the minimal polynomial $m_{α}$ . Consequently the number of embeddings is the number of distinct roots of $m_{α}$ , not the sum of their multiplicities.

Facts & Assumptions

Given: An algebraic element $α$ over $F$ , its minimal polynomial $m_{α}$ , and an algebraically closed overfield $Ω$ of $F$ .

[L1]

An $F$ -embedding carries an algebraic element to a conjugate root of its minimal polynomial (A base-field embedding carries an algebraic element to a conjugate).

[L2]

For a monic irreducible polynomial, every chosen root in an extension induces a unique homomorphism from the quotient adjoining that root (Universal property of adjoining a root of an irreducible polynomial).

[L3]

Every nonconstant polynomial over an algebraically closed field has a root there (An algebraically closed field: every nonconstant polynomial has a root in the field).

Proof

technique · direct

1.1

By [L1], the image $σ (α)$ of every $F$ -embedding is a root of $m_{α}$ in $Ω$ .

1.2

Conversely, if $β \in Ω$ is a root of $m_{α}$ , [L2] applied to the two realizations of $F [x] / (m_{α})$ gives a unique $F$ -embedding $F (α) \to Ω$ with $α \mapsto β$ .

2.1

The constructions in steps 1.1 and 1.2 are inverse because an $F$ -homomorphism on $F (α)$ is determined by the image of $α$ .

step 1.1step 1.2

3.1

The polynomial $m_{α}$ splits in $Ω$ by repeated use of [L3], and the bijection indexes embeddings by its distinct roots, so repeated roots are counted once.

step 2.1L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 31 results over 7 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

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)

FF-embeddings of F(α)F(α) into an algebraically closed field correspond to the distinct roots of mαmα​