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.
Agreement of the Liouville and minimum-modulus proofs of the fundamental theorem of algebra
Remark
The theorem Fundamental theorem of algebra by Liouville's theorem applies Liouville's theorem to the reciprocal of a hypothetical zero-free polynomial. The theorem Fundamental theorem of algebra: every nonconstant complex polynomial has a complex root establishes the same root-existence statement by descending from a positive minimum of the polynomial's modulus. The routes are independent: the Liouville argument does not cite the minimum-modulus theorem, and the minimum-modulus argument does not use Liouville's theorem.
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Used by
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Dependency tree · two levels
11 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
- Lars Ahlfors, Complex Analysis, 3rd ed., Ch. 4 §2.3 (standard reference, not scraped)
- E. Stein and R. Shakarchi, Complex Analysis, Ch. 2, Corollary 4.6 (standard reference, not scraped)
- Matthias Weber, Complex Analysis, Corollary 2.3.3 (standard reference, not scraped)
- Steven G. Krantz, A Guide to Complex Variables, §3.1.4 (standard reference, not scraped)