PropositionStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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.

The relative algebraic closure of $F$ in $K$ has no further algebraic elements inside $K$

Statement

Let $A = {acl}_{K} (F)$ . If $b \in K$ is algebraic over $A$ , then $b \in A$ . Equivalently, $A$ is relatively algebraically closed in $K$ .

Facts & Assumptions

Given: A field extension $K / F$ , its relative algebraic closure $A$ , and an element $b \in K$ algebraic over $A$ .

[L1]

The field $A$ consists exactly of the elements of $K$ algebraic over $F$ (The relative algebraic closure of $F$ in an extension $K$ ).

[L2]

Algebraicity is transitive in towers of field extensions (Algebraicity is transitive in towers of field extensions).

[L3]

An algebraic element generates a finite simple extension (An element is algebraic over $F$ if and only if its simple extension $F (a) / F$ is finite).

[L4]

Every finite extension is algebraic (Every finite field extension is algebraic).

Proof

technique · direct

1.1L1

By [L1], every element of $A$ is algebraic over $F$ , so $A / F$ is algebraic.

2.1givenstep 1.1L2L3L4

The element $b$ is algebraic over $A$ by hypothesis, so [L3] makes $A (b) / A$ finite and [L4] makes it algebraic. Applying [L2] to $F \subseteq A \subseteq A (b)$ shows that $b$ is algebraic over $F$ .

3.1step 2.1L1∎

By [L1], an element of $K$ algebraic over $F$ belongs to $A$ , hence $b \in A$ .

Depends on

Used by

Nothing in the library uses this result yet.

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

A. W. Knapp, Basic Algebra, 2nd ed., Chapter IX, Section 1 (standard reference, not scraped)

The relative algebraic closure of FF in KK has no further algebraic elements inside KK