Alphabeta MathHow 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
- x⁵-6x+3 over ℚ is not solvable by radicals Example
- Separable generation after finite purely inseparable extensions Lemma
- 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
- Dense regular loci on every component Theorem
- Jacobian criterion and openness of the regular locus over a perfect field Theorem
- Openness of the regular locus over a perfect field Theorem
- Over a perfect field, every endomorphism has a unique commuting semisimple-plus-nilpotent decomposition, polynomial in the endomorphism Theorem
- Pure inseparability and its conjugate, embedding, and separable-degree criteria Theorem
- Regular algebras over a perfect field are geometrically regular Theorem
Dependency tree · two levels
6 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
- 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)