Fundamental Theorem of Algebra¶

Theorem¶

The fundamental theorem of algebra states that every nonconstant polynomial with complex number coefficients has at least one root in $\mathbb{C}$ .

Interpretation¶

The theorem says that $\mathbb{C}$ is large enough to solve all nonconstant single-variable polynomial equations over $\mathbb{C}$ . In $\mathbb{R}$ , by contrast, $x^2+1=0$ has no solution.

The five fifth roots of unity as equally spaced points on the unit circle, 2 pi over 5 apart, with only the root 1 on the real axis.

The figure shows the roots of $z^5-1$ $1$ , that is, the solutions of $z^5=1$ $1$ . By De Moivre's formula a solution must have modulus $1$ , since $|z|^5=1$ $1$ , and an angle $\varphi$ for which $5\varphi$ is a multiple of $2\pi$ . This gives five points on the unit circle, $2\pi/5$ apart. Only $z=1$ is real: over $\mathbb{R}$ the polynomial has one root, over $\mathbb{C}$ it has five, as many as its degree.

Change the coefficients, even to complex ones: a polynomial of degree $n$ always has exactly $n$ roots in $\mathbb{C}$ , counted with multiplicity.

The theorem guarantees only one root, but that is enough to find all of them. If $\lambda$ is a root, polynomial long division by $z-\lambda$ leaves remainder $0$ and a quotient of one degree lower, which again has a root if it is nonconstant. Repeating the step splits a polynomial of degree $n\ge1$ into $n$ linear factors, so it has exactly $n$ roots counted with multiplicity; $x^3$ has the single root $0$ , counted three times.

The word nonconstant is needed, since a nonzero constant polynomial has no root. The theorem says that roots exist, not how to compute them. For polynomials with real coefficients the non-real roots come in pairs by the complex conjugate root theorem, and in the language of fields the theorem says that $\mathbb{C}$ is algebraically closed.

Key Results¶

Every complex polynomial of degree $n\geq1$ has exactly $n$ roots when counted with multiplicity.
Every complex polynomial of degree $n\geq1$ can be factored into linear factors over $\mathbb{C}$ .

Examples¶

$x^2+1$ has roots $\mathrm{i}$ and $-\mathrm{i}$ in $\mathbb{C}$ .
$x^3$ has the root $0$ with multiplicity $3$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 1

Aliases: FTA, d'Alembert-Gauss theorem, d'Alembert's theorem Symbols: complex numbers, lambda, phi, pi, real numbers

Last updated: 20 October 2026 03:02