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 Artin and minimum-modulus proofs of the fundamental theorem of algebra use different machinery
This page proves the fundamental theorem of algebra by the Artin route: one use
of the intermediate value theorem for odd-degree real polynomials, followed by a
Galois-theoretic argument using Sylow theory and quadratic extensions. A later
item, thm-fundamental-theorem-of-algebra-minimum-modulus-proof, proves the
same root-existence statement by a different route, using the minimum-modulus
method from complex analysis. Neither proof cites the other.
Milne's honesty note applies here as well: this is not purely a theorem of algebra. The present proof keeps the analytic input small: it uses the intermediate value theorem for the odd-degree root argument and the real square-root existence behind the published complex square-root theorem, while the later minimum-modulus proof spends much more analytic machinery.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
16 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
- J. S. Milne, Fields and Galois Theory, v5.10, Chapter 5 (standard reference, not scraped)
- J. Lebl, Basic Analysis I, The Fundamental Theorem of Algebra (standard reference, not scraped)