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.
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- 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.
An algebraic closure of a field
Definition
An algebraic closure of a field is a field extension that is algebraic (Algebraic and transcendental elements and algebraic extensions) and whose field is algebraically closed (An algebraically closed field: every nonconstant polynomial has a root in the field). The notation denotes a chosen algebraic closure; it does not specify a preferred one or a preferred isomorphism between two choices.
Depends on
Used by
- Assuming Choice, any two algebraic closures are base-isomorphic Corollary
- Assuming Choice, conjugates in an algebraic closure are related by a base automorphism Corollary
- Global functions on geometrically connected and geometrically reduced proper schemes Corollary
- The algebraic numbers in ℂ form an algebraic closure of ℚ Corollary
- The complex numbers form an algebraic closure of ℝ Corollary
- Geometric fibres and geometric points Definition
- Semisimple endomorphisms as endomorphisms diagonalisable over an algebraic closure, and nilpotent endomorphisms Definition
- The normal closure of an algebraic extension inside a fixed algebraic closure Definition
- The separable degree [K:F]ₛ as a count of embeddings into an algebraic closure Definition
- A binary resultant detects a common root at infinity lost by naive dehomogenization Example
- Fₚ is the union of its finite subfields and is an infinite algebraic extension Example
- Resultant of two binary linear forms Example
- FALSE: an algebraic closure is unique up to a unique base-field isomorphism False statement
- Flat maps with geometrically regular fibres have standard smooth local presentations Lemma
- Assuming Choice, every field has an algebraic closure Theorem
- Global functions on proper integral schemes form a finite extension of the base field Theorem
- Over a perfect field, every endomorphism has a unique commuting semisimple-plus-nilpotent decomposition, polynomial in the endomorphism Theorem
- The binary Sylvester resultant detects a common geometric projective root 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 and 5 (standard reference, not scraped)
- J. S. Milne, Fields and Galois Theory, Chapter 6 (standard reference, not scraped)