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.
Separable algebraic elements and separable extensions
Definition
Let be a field extension. An element is separable over when it is algebraic over (Algebraic and transcendental elements and algebraic extensions) and its minimal polynomial over (The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element) is a separable polynomial (Repeated roots in extension fields and separable polynomials). The extension is separable when every element of is separable over .
Depends on
Used by
- An extension that is both separable and purely inseparable is trivial Corollary
- Every algebraic extension of a perfect field is separable Corollary
- A finite extension generated by elements all but possibly one of which are separable is simple Theorem
- A finite extension is separable if and only if [K:F]ₛ=[K:F] Theorem
- An algebraic extension generated by separable elements is separable Theorem
- Separability is transitive in towers of algebraic extensions Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 27 results over 7 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)