CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 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.

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Assuming Choice, any two algebraic closures are base-isomorphic

Statement

Assuming the Axiom of Choice, any two algebraic closures of a field $F$ are $F$ -isomorphic. No uniqueness of the isomorphism is asserted.

Facts & Assumptions

Given: The Axiom of Choice and two algebraic closures $Ω_{1} / F$ and $Ω_{2} / F$ .

[L1]

Assuming Choice, a base embedding into an algebraically closed field extends across an algebraic extension (Assuming Choice, a base-field embedding extends across every algebraic extension).

[L2]

An algebraic closure is algebraic over its base and algebraically closed (An algebraic closure of a field).

[L3]

Every algebraic element has a monic irreducible minimal polynomial over the base (The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element).

Proof

technique · direct

1.1L1L2

Extend the identity embedding of $F$ across $Ω_{1} / F$ into $Ω_{2}$ by [L1], obtaining an $F$ -embedding $σ : Ω_{1} \to Ω_{2}$ .

2.1step 1.1L2L3

Its image $E = σ (Ω_{1})$ is algebraically closed because it is isomorphic to $Ω_{1}$ . Every $b \in Ω_{2}$ is algebraic over $F \subseteq E$ , so [L3] gives a minimal polynomial over $E$ ; this polynomial has a root in $E$ , and irreducibility then makes it linear. Hence $b \in E$ .

3.1step 1.1step 2.1∎

Thus $σ$ is surjective as well as injective, and is an $F$ -isomorphism. The argument proves existence only and makes no uniqueness assertion.

Depends on

Used by

FALSE: an algebraic closure is unique up to a unique base-field isomorphism False statement
Independence of the chosen algebraic closure Lemma

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

J. S. Milne, Fields and Galois Theory, Chapter 6 (standard reference, not scraped)
P. L. Clark, Field Theory, Chapter 4 (standard reference, not scraped)