Fundamental Theorem of Algebra

The fundamental theorem of algebra states that every non-constant polynomial with complex coefficients has at least one complex root. This remarkable result shows that $\mathbb{C}$ is algebraically closed.

Statement

Theorem1.3Fundamental theorem of algebra

Every non-constant polynomial $p(z) = a_n z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0$ with complex coefficients and $a_n \neq 0$ has at least one root in $\mathbb{C}$ . That is, there exists $z_0 \in \mathbb{C}$ such that $p(z_0) = 0$ .

RemarkFactorization consequence

By induction, every polynomial of degree $n \geq 1$ factors completely into linear factors:

$p(z) = a_n (z - z_1)(z - z_2) \cdots (z - z_n)$

where $z_1, \ldots, z_n \in \mathbb{C}$ (counted with multiplicity). This is why $\mathbb{C}$ is called algebraically closed: every polynomial equation has all its roots within $\mathbb{C}$ .

Proof Strategy

RemarkProofs

There are many proofs of the fundamental theorem, using tools from:

Complex analysis: If $p$ has no zeros, then $1/p(z)$ is entire and bounded (since $|p(z)| \to \infty$ as $|z| \to \infty$ ). By Liouville's theorem, $1/p$ is constant, a contradiction.
Topology: Using the fact that $p: \mathbb{C} \to \mathbb{C}$ induces a nontrivial map on fundamental groups if $p$ has no zeros.
Real analysis: Showing that $|p(z)|$ attains a minimum at some $z_0 \in \mathbb{C}$ , and proving that this minimum must be zero.

The complex-analytic proof is elegant but requires Liouville's theorem. A more elementary proof is given in Proof 1.

Examples and Applications

ExampleQuadratic polynomials

Every quadratic $p(z) = az^2 + bz + c$ (with $a \neq 0$ ) has two roots (counted with multiplicity):

$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$

The discriminant $\Delta = b^2 - 4ac$ can be any complex number. Unlike in real analysis, there is no distinction between "positive" and "negative" discriminants — every quadratic has two complex roots.

ExampleCyclotomic polynomials

The polynomial $z^n - 1$ factors as

$z^n - 1 = (z - \omega_0)(z - \omega_1) \cdots (z - \omega_{n-1})$

where $\omega_k = e^{2\pi i k/n}$ are the $n$ -th roots of unity. This is a direct application of the fundamental theorem.

ExamplePolynomials with no real roots

$p(z) = z^2 + 1$ has no real roots, but has two complex roots: $z = i$ and $z = -i$ . Thus

$z^2 + 1 = (z - i)(z + i).$

More generally, every real polynomial of odd degree has at least one real root (by the intermediate value theorem), but polynomials of even degree need not have real roots.

Historical Remark

RemarkHistory

The fundamental theorem was first stated by d'Alembert (1746) and proved rigorously by Gauss (1799), who gave four different proofs over his lifetime. The name "fundamental theorem of algebra" is somewhat misleading: all rigorous proofs use tools from analysis or topology, not just algebra. A purely algebraic proof is impossible in first-order logic, as shown by model theory.

The theorem is fundamental because it guarantees that polynomial equations always have solutions in $\mathbb{C}$ , making $\mathbb{C}$ the "correct" number system for algebra.

Relation to Other Results

RemarkConnection to Liouville's theorem

The most elegant proof uses Liouville's theorem: every bounded entire function is constant. If $p(z)$ is a non-constant polynomial with no zeros, then $f(z) = 1/p(z)$ is entire (holomorphic on all of $\mathbb{C}$ ). Since $|p(z)| \to \infty$ as $|z| \to \infty$ , we have $|f(z)| \to 0$ , so $f$ is bounded. By Liouville, $f$ is constant, implying $p$ is constant — a contradiction.

This proof is given in detail in Liouville's theorem.

Summary

RemarkKey points

Every non-constant polynomial $p(z)$ has at least one complex root.
As a consequence, $p(z)$ factors completely into linear factors over $\mathbb{C}$ .
$\mathbb{C}$ is algebraically closed: polynomial equations are always solvable.
The theorem requires tools beyond pure algebra — complex analysis provides the most elegant proofs.
The fundamental theorem underlies much of complex analysis, including the theory of entire functions and meromorphic functions.