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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 .
A nonconstant complex polynomial tends to infinite modulus and attains a global minimum modulus states that attains a global minimum on .
A nonzero value of a nonconstant complex polynomial cannot be a local minimum of its modulus states that, if , then is not minimal on any neighbourhood of .
Proof
By [L1], let minimize globally.
Suppose .
By [L2], there is a point arbitrarily near with strictly smaller modulus, contradicting step 1.1.
Therefore , proving the theorem.
Depends on
Used by
- Certified complex-polynomial root isolation in exact algebraic arithmetic Algorithm
- Real bounded-polynomial QSP phase synthesis Algorithm
- Bezout needs algebraic closure: an imaginary conic has no real point Counterexample
- Analytic primary matrix functions by Jordan blocks Definition
- FALSE: every entire function with an antiderivative is a polynomial False statement
- A nonnegative even polynomial has a bounded parity complement Lemma
- Additive characters are exactly one-dimensional complex representation characters Lemma
- Engel, the trace criterion, and Killing nondegeneracy Lemma
- Every regular complex pencil has a homogeneous eigenvalue pair Lemma
- Exact interval boundedness by multiplicity strata and Sturm counts Lemma
- Finite Lie triangularization and rank-one complete reducibility Lemma
- Finite semisimple Cartan, root and string structure Lemma
- Nonzero finite dimensional complex invariant subspaces have unitary eigenvectors Lemma
- Real-root intervals certify the sign of a real polynomial Lemma
- The normalized adjoint Casimir is a positive scalar Lemma
- The root condition is equivalent to bounded powers of the multistep companion matrix Lemma
- Agreement of the Liouville and minimum-modulus proofs of the fundamental theorem of algebra Remark
- The fundamental-group and minimum-modulus proofs of the fundamental theorem of algebra Remark
- Correctness and termination of certified complex-polynomial root isolation Theorem
- Jordan, contour, similarity, and normal-functional-calculus formulas agree Theorem
Dependency tree · two levels
10 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. Lebl, Basic Analysis I: The Fundamental Theorem of Algebra (standard reference, not scraped)