Alphabeta MathDefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17
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.
Perfect fields: every irreducible polynomial is separable
Definition
A field (Field) is perfect when every nonconstant irreducible polynomial in is separable (Repeated roots in extension fields and separable polynomials).
Depends on
Used by
- Every algebraic extension of a perfect field is separable Corollary
- A field is perfect exactly when it has characteristic zero or its Frobenius map is surjective Theorem
- An algebraic extension is purely inseparable over its separable closure Theorem
- Pure inseparability and its conjugate, embedding, and separable-degree criteria Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 19 results over 6 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
- P. L. Clark, Field Theory, Chapters 3 to 5 (standard reference, not scraped)
- J. S. Milne, Fields and Galois Theory, Chapters 2, 3, and 5 (standard reference, not scraped)