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The 41 pages that link to Fundamental theorem of algebra, each with the reason it gives.

• Complex numberRelated: Complex numbers provide the setting in which polynomial equations always have roots.

• Carl Friedrich GaussRelated: Gauss supplied the first widely accepted proof and returned to the theorem repeatedly.

• Algebraic numberRelated: It establishes the complex setting in which algebraic numbers are defined.

• Cauchy's integral formulaRelated: One proof uses Cauchy's formula and the consequences of holomorphicity to rule out a rootless polynomial.

• Fundamental groupRelated: A winding-number argument using the fundamental group proves a standard form of it.

• Maximum modulus principleRelated: A standard proof uses modulus minimization and the local structure of a polynomial near a nonzero minimum.

• Polynomial factorizationRelated: It guarantees complete linear factorization over the complex numbers, unlike over the rationals.

• Liouville's theoremRelated: A standard proof assumes a rootless polynomial's reciprocal is bounded and entire, contradicting Liouville's theorem.

• Polynomial equationRelated: It guarantees the total number of complex roots when multiplicity is counted.

• Polynomial degreeRelated: A degree-n polynomial has exactly n complex roots when multiplicities are counted.

• Polynomial functionRelated: It guarantees that a degree-n polynomial has exactly n complex roots when multiplicity is counted.

• Argument principleRelated: The argument principle provides a complex-analysis route to counting polynomial roots.

• Joseph LiouvilleRelated: Liouville’s complex-analysis theorem offers a classic route to proving this result.

• Cubic equationRelated: It guarantees three complex roots for every cubic, counting multiplicity.

• Rational root theoremRelated: It guarantees complex roots exist, while the rational root theorem only constrains rational ones.

• Vieta's formulasRelated: Factoring into linear terms provides the roots from which the formulas recover coefficients.

• Algebraically closed fieldRelated: It establishes that the complex numbers satisfy the defining root condition.

• History of algebraRelated: Its proof history exposed changing standards for what counted as a rigorous algebraic argument.

• Partial fraction decompositionRelated: Over the complex numbers, it guarantees that denominator factors can all be linear.

• Rouché's theoremRelated: Rouché's theorem can locate how many of a polynomial's roots lie in a chosen region.

• Zero of a functionRelated: It guarantees that complex zeros exist for every nonconstant polynomial.

• Descartes' rule of signsRelated: The rule addresses real roots within the broader theory of polynomial roots.

• Sturm's theoremRelated: Sturm's theorem addresses the real-root information that the fundamental theorem does not provide.

• Gauss–Lucas theoremRelated: Its factorization into roots supplies the representation used in the theorem’s proof.

• Complex conjugate root theoremRelated: It guarantees a full set of complex roots, while conjugation constrains their arrangement for real polynomials.

• Casas-Alvero conjectureRelated: It guarantees a root for the polynomial, but not that the same root works for all derivatives.

• Gelfand–Mazur theoremRelated: Its algebraic closure is reflected in the spectral argument behind the theorem.

• Marden's theoremRelated: It guarantees a cubic's three roots when multiplicities are counted.