TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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Fundamental theorem of algebra: every nonconstant complex polynomial has a complex root

Statement

Every nonconstant complex polynomial has a complex root. The conventions and prerequisite facts used below are recorded in A nonconstant complex polynomial tends to infinite modulus and attains a global minimum modulus, A nonzero value of a nonconstant complex polynomial cannot be a local minimum of its modulus.

Facts & Assumptions

Given: A nonconstant complex polynomial $p$ .

[L1]

A nonconstant complex polynomial tends to infinite modulus and attains a global minimum modulus states that $|p|$ attains a global minimum on $\mathbb C$ .

[L2]

A nonzero value of a nonconstant complex polynomial cannot be a local minimum of its modulus states that, if $p(a)\ne0$ , then $|p|$ is not minimal on any neighbourhood of $a$ .

Proof

technique · contradiction

1.1

By [L1], let $a$ minimize $|p|$ globally.

1.2

Suppose $p(a)\ne0$ .

assume-contra

2.1

By [L2], there is a point arbitrarily near $a$ with strictly smaller modulus, contradicting step 1.1.

L2step 1.1

3.1

Therefore $p(a)=0$ , proving the theorem.

discharge-contradiction∎

Depends on

Used by

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Sources

J. Lebl, Basic Analysis I: The Fundamental Theorem of Algebra (standard reference, not scraped)