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.
The -th roots of a complex number and the distinct roots of unity for every
Statement
Write for the canonical-natural map of The canonical natural of a field. If with and , its distinct roots are For , the only th root is . Thus the th roots of unity are precisely for with . The conventions and prerequisite facts used below are recorded in Every nonzero complex number has a unique polar form with and , Complex de Moivre formula for every integer exponent, , and exactly when , Existence and uniqueness of -th roots: a unique with , The canonical natural of a field, Integer powers in the complex field.
Facts & Assumptions
Given: with and .
Proof
For , construct each listed candidate from the positive real th root of ; de Moivre verifies it.
The kernel theorem shows two listed candidates coincide only when their indices are equal modulo .
Conversely polar form and the same kernel calculation force every root onto the list; the case is immediate.