Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13
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 field of fractions of an integral domain
Definition
If is an integral domain, then is multiplicative. Its localisation is the field of fractions of . Thus its elements are fractions with and , modulo the localisation equivalence relation.
Depends on
Used by
- For a field F, F(t)=Frac(F[t]) is its rational function field; in particular ℝ(t)=Frac(ℝ[t]) Corollary
- Outside a domain, the nonzero elements need not be multiplicative: 2·3=0 in ℤ/6 Counterexample
- Frac(ℤ) is canonically isomorphic to ℚ Example
- Frac(D) is a field and d↦ d/1 embeds the integral domain D Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 18 results over 12 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
- The CRing Project, Chapter 13: Fields and Extensions (standard reference, not scraped)