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-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.
Fields of characteristic zero, finite fields, and algebraically closed fields are perfect
Statement
Every field of characteristic zero is perfect. Every finite field is perfect, and every algebraically closed field is perfect.
Facts & Assumptions
Given: A field in one of the classes named in the Statement.
Perfectness is equivalent to characteristic zero or, in characteristic , surjectivity of Frobenius (A field is perfect exactly when it has characteristic zero or its Frobenius map is surjective).
Frobenius is an automorphism of every finite field (Frobenius is an injective endomorphism in characteristic , and an automorphism for finite fields).
In an algebraically closed field, every nonconstant polynomial has a root (An algebraically closed field: every nonconstant polynomial has a root in the field).
Proof
The characteristic-zero case is immediate from [L1].
If is finite of characteristic , [L2] makes Frobenius surjective, so [L1] makes perfect.
If is algebraically closed of characteristic , then for every the polynomial has a root by [L3]; hence every is a th power and [L1] makes perfect.