Existing in Mathlib · Algebra and complex analysis

Fundamental Theorem of Algebra

Take a polynomial whose coefficients and inputs are complex numbers. If its degree is positive, then at least one complex number makes the polynomial equal to zero.

complex numbers
polynomials
roots
degree

Exact theorem

Exact Mathlib statement

theorem Complex.exists_root {f : Polynomial ℂ} (hf : 0 < f.degree) : ∃ z, f.IsRoot z

The theorem at a glance

Fundamental Theorem of Algebra at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory theorem poster moves from the positive-degree condition for a complex polynomial to one luminous complex root, with a four-step reading guide and an exact-scope boundary. — Positive degree leads to the existence of at least one complex root; the poster visibly limits the claim to root existence.

Statement structure

Statement and scope

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

Scientific figure illustrating: A positive-degree complex polynomial is pictured descending to a root where it vanishes. — A positive-degree complex polynomial is pictured descending to a root where it vanishes.

Why it matters

A mathematical landmark

The theorem says that complex numbers contain the roots needed by every nonconstant one-variable complex polynomial. It is a foundational bridge between algebra and complex analysis.

Proof architecture

A Liouville proof of root existence

4 curated stages

Mathlib proves the exact root-existence theorem by assuming a positive-degree complex polynomial has no root, studying its differentiable reciprocal, using polynomial growth and Liouville's theorem to force that reciprocal to vanish everywhere, and deriving the impossible conclusion that the polynomial is the zero constant.

These source-anchored stages explain the retained Mathlib proof route. They are generated explanations, not proof evidence or an extracted Lean proof-term dependency graph.

Assume the positive-degree polynomial has no root
Assume the positive-degree polynomial has no root
Assume for contradiction that the positive-degree complex polynomial never vanishes, so its evaluation ribbon avoids the root socket everywhere.
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Assume for contradiction that the positive-degree complex polynomial never vanishes, so its evaluation ribbon avoids the root socket everywhere.
0 < f.degree and ∀ z : ℂ, f.eval z ≠ 0
The proof begins by negating the desired root existence. Under this contradiction hypothesis, f.eval z is nonzero for every complex z, which supplies the pointwise nonvanishing condition needed to differentiate its reciprocal.
Reciprocal decay follows polynomial growth
Reciprocal decay follows polynomial growth
With no zero to obstruct inversion, the reciprocal is differentiable; polynomial growth at infinity becomes reciprocal decay to zero.
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With no zero to obstruct inversion, the reciprocal is differentiable; polynomial growth at infinity becomes reciprocal decay to zero.
‖f(z)‖ → ∞ and (f.eval z)⁻¹ → 0 as z tends to complex infinity
Because f.eval is everywhere nonzero, its reciprocal is differentiable. Positive degree makes the norm of the polynomial tend to infinity along the complex cobounded filter, and inversion turns that growth into convergence of the reciprocal to zero.
Liouville forces the reciprocal to vanish everywhere
Liouville forces the reciprocal to vanish everywhere
A differentiable reciprocal that tends to zero at complex infinity is forced by Liouville's theorem to equal zero at every point.
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A differentiable reciprocal that tends to zero at complex infinity is forced by Liouville's theorem to equal zero at every point.
∀ z : ℂ, (f.eval z)⁻¹ = 0
Mathlib applies the Liouville consequence apply_eq_of_tendsto_cocompact to the differentiable reciprocal and its limit at complex infinity. The conclusion is pointwise and global: for every complex z, the reciprocal of f.eval z equals zero.
The zero polynomial contradiction reveals a root
The zero polynomial contradiction reveals a root
Pointwise zero reciprocals force the polynomial to be the zero constant, contradicting positive degree; therefore a complex root exists.
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Pointwise zero reciprocals force the polynomial to be the zero constant, contradicting positive degree; therefore a complex root exists.
f = C 0, contradicting 0 < f.degree; hence ∃ z : ℂ, f.IsRoot z
Inverse injectivity and polynomial extensionality turn the pointwise reciprocal equation into f = C 0. That makes the polynomial's degree incompatible with the original positive-degree hypothesis, so the rootless assumption is false and a complex root exists.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib's exact existence theorem for a root of a positive-degree complex polynomial. It does not by itself state complete linear factorization, count roots with multiplicity, claim a new proof, or transfer any accepted-result status.

Proof Atlas did not originate the Fundamental Theorem of Algebra or Mathlib's declaration.
The direct Mathlib evidence is a local reproduction of an upstream theorem, not a new formalization.
The generated explanation and visual are not proof evidence.

Source and local evidence

Where the theorem comes from

Existing declaration: Complex.exists_root in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.fundamental-theorem-algebra.v001
Source: Open the pinned upstream reference