RemarkRemark: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

How 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-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Rouche gives the standard leading-term proof of the fundamental theorem of algebra

Remark

Let

$p (z) = a_{n} z^{n} + a_{n - 1} z^{n - 1} + \dots + a_{0}, a_{n} \neq 0.$

On the circle $∣ z ∣ = R$ one has

$∣ a_{n - 1} z^{n - 1} + \dots + a_{0} ∣ \leq ∣ a_{n - 1} ∣ R^{n - 1} + \dots + ∣ a_{0} ∣,$

so for sufficiently large $R$ the lower-degree tail is strictly smaller than $∣ a_{n} ∣ R^{n} = ∣ a_{n} z^{n} ∣$ . Rouché's theorem therefore gives the same number of zeros for $p$ and its leading term $a_{n} z^{n}$ inside $∣ z ∣ < R$ , namely $n$ counting multiplicity.

This is exactly the standard leading-term proof that every nonconstant complex polynomial has roots and, more sharply, that a degree- $n$ polynomial has exactly $n$ roots counted with multiplicity, agreeing with the canonical result A complex polynomial of degree $n$ has exactly $n$ roots counted with multiplicity.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

R. W. Howell and J. H. Mathews, Complex Analysis, §8.7 (standard reference, not scraped)
J. Lebl, Guide to Cultivating Complex Analysis, §5.4 (standard reference, not scraped)