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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An algebraic extension containing a root of every nonconstant base polynomial is algebraically closed

Statement

Let $L / F$ be algebraic. If every nonconstant polynomial in $F [x]$ has a root in $L$ , then $L$ is algebraically closed. One root-adjoining extension suffices; no iterated tower of root extensions is required.

Facts & Assumptions

Given: An algebraic extension $L / F$ containing a root of every nonconstant polynomial over $F$ .

[L1]

The one-step root condition over a perfect base makes an algebraic extension algebraically closed (The one-step root condition makes an algebraic extension of a perfect field algebraically closed).

[L2]

In positive characteristic, the elements with a suitable $p$ -power in the base form a perfect intermediate field whose polynomials retain the root condition in $L$ (The elements with a $p^{n}$ th power in the base form a perfect subfield carrying the one-step root condition).

[L3]

Every characteristic-zero field is perfect (Fields of characteristic zero, finite fields, and algebraically closed fields are perfect).

Proof

technique · direct

1.1

If $F$ has characteristic zero, [L3] makes it perfect and [L1] makes $L$ algebraically closed.

L1L3

1.2

If $F$ has characteristic $p > 0$ , let $F^{'}$ be the perfect intermediate field from [L2]. The extension $L / F^{'}$ is algebraic because $L / F$ is algebraic, and [L2] gives the one-step root condition over $F^{'}$ , so [L1] again makes $L$ algebraically closed.

L1L2

2.1

The characteristic-zero and positive-characteristic cases exhaust all fields and establish the conclusion without repeating the root-extension construction.

step 1.1step 1.2∎

Depends on

Used by

Assuming Choice, every field has an algebraic closure Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 48 results over 10 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

J. S. Milne, Fields and Galois Theory, Proposition 6.5 (standard reference, not scraped)
P. L. Clark, Field Theory, Theorem 4.9 (standard reference, not scraped)