DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

The prime subfield as the intersection of all subfields

Definition

Let $F$ be a field. Its prime subfield is

$F_{0} : = ⋂ {E : E is a subfield of F} .$

The family is nonempty because it contains $F$ . Its intersection is a subfield: every member contains $0$ and $1$ and is closed under subtraction, multiplication, and inversion of nonzero elements, so the intersection has the same properties. It is the unique smallest subfield of $F$ .

Depends on

Used by

A field's prime subfield is isomorphic to ℚ in characteristic zero and to Fₚ in characteristic p Theorem

Dependency tree · two levels

9 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 3 (standard reference, not scraped)