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.
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
- 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
- overlineFₚ is the union of its finite subfields and is an infinite algebraic extension Example
- FALSE: an algebraic closure is unique up to a unique base-field isomorphism False statement
- Assuming Choice, every field has an algebraic closure Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 11 results over 4 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 and 5 (standard reference, not scraped)
- J. S. Milne, Fields and Galois Theory, Chapter 6 (standard reference, not scraped)