DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finitely generated field extensions $F (a_{1}, \dots, a_{r})$

Definition

Let $K / F$ be a field extension and let $a_{1}, \dots, a_{r} \in K$ . The notation

$F (a_{1}, \dots, a_{r})$

means the smallest subfield of $K$ containing $F$ and all the $a_{i}$ . An extension $K / F$ is finitely generated when $K = F (a_{1}, \dots, a_{r})$ for some finite list. For the empty list, $F () = F$ .

Depends on

Used by

An extension generated by finitely many algebraic elements is finite Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 40 results over 11 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

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

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