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.
Assuming Choice, every field has an algebraic closure
Statement
Assuming the Axiom of Choice, every field has an algebraic closure.
Facts & Assumptions
Given: A field and the Axiom of Choice.
Assuming Choice, there is an algebraic extension containing a root of every nonconstant polynomial over (Assuming Choice, every field has an algebraic extension containing roots of all nonconstant base polynomials).
Every algebraic extension with that one-step root property is algebraically closed (An algebraic extension containing a root of every nonconstant base polynomial is algebraically closed).
An algebraic closure is an algebraic extension that is algebraically closed (An algebraic closure of a field).
Proof
Use [L1] to construct an algebraic extension containing a root of every nonconstant base polynomial.
By [L2], this same field is already algebraically closed.
Thus is an algebraic closure by [L3].