TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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

1.1L1L3

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

1.2L1L2

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.

2.1step 1.1step 1.2∎

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

Depends on

Used by

Assuming Choice, every field has an algebraic closure Theorem

Dependency tree · two levels

13 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, Proposition 6.5 (standard reference, not scraped)
P. L. Clark, Field Theory, Theorem 4.9 (standard reference, not scraped)