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.
Purely inseparable algebraic extensions
Definition
Let be algebraic (Algebraic and transcendental elements and algebraic extensions). If , the extension is purely inseparable when for every there is such that . The exponent is allowed. If , the term purely inseparable is reserved for the trivial extension .
The powers in positive characteristic are governed by the Frobenius endomorphism of Frobenius is an injective endomorphism in characteristic , and an automorphism for finite fields.
Depends on
Used by
- Global functions on geometrically connected and geometrically reduced proper schemes Corollary
- A regular point that is not smooth: a purely inseparable thickening Counterexample
- Fₚ(t)/Fₚ(tᵖ) is normal and inseparable with trivial automorphism group Counterexample
- Regular field factors can have a nonregular tensor product Counterexample
- p-bases for finite exponent-one purely inseparable extensions Definition
- A finite normal extension is separable over its purely inseparable fixed field Lemma
- Finite purely inseparable rational extensions admit a finite Frobenius envelope Lemma
- Integral closure in a purely inseparable rational envelope is finite Lemma
- An algebraic extension is purely inseparable over its separable closure Theorem
- Pure inseparability and its conjugate, embedding, and separable-degree criteria 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
- P. L. Clark, Field Theory, Chapters 4 and 5 (standard reference, not scraped)
- J. S. Milne, Fields and Galois Theory, Chapters 3 and 5 (standard reference, not scraped)