The statement that every nonconstant polynomial with complex coefficients has at least one complex root; equivalently, every degree-n polynomial over the complex numbers factors as a product of n linear factors when multiplicity is counted.

Principle

The algebraic closure of the complex field ensures that polynomial equations cannot avoid roots: completeness of the complex numbers and algebraic properties force factorisation into linear factors.

Demonstration

Example: the polynomial p(z)=z^2+1 has the complex roots i and −i, and the cubic z^3−1 factors as (z−1)(z−e^{2πi/3})(z−e^{4πi/3}), illustrating existence of complex roots and full factorisation.

Misapplication

Applying the theorem over the real numbers to conclude every real polynomial has a real root; for instance x^2+1 has no real root. Another misuse is asserting an explicit closed-form expression for arbitrary roots when no algebraic formula exists.

Consequence

Any polynomial over C can be reduced to linear factors, which yields existence of eigenvalues of complex matrices, complete factorisation, and many structural results in algebra and analysis.

Reversal

Over a non-algebraically-closed field such as R, nonconstant polynomials can lack roots; reversing the claim highlights that algebraic closure is the essential hypothesis rather than a generic property.

Boundary

Applies to polynomials with complex coefficients and positive degree. It does not apply to constant polynomials, to polynomials over fields that are not algebraically closed, nor does it guarantee simple constructive formulas for roots in general.

Semantic Tension

The theorem guarantees existence but says nothing about computational solvability or closed-form expressibility of roots; there is tension between existence results and effective algorithms or formulas.

Synthesis

The Fundamental Theorem of Algebra identifies the complex numbers as algebraically closed for single-variable polynomials: existence of at least one root implies full linear factorisation and underpins many results in algebra and linear analysis.