Algebraically Closed Field¶

Definition¶

A field $\mathbb{F}$ is algebraically closed if every nonconstant polynomial in $\mathbb{F}[x]$ (that is, with coefficients in $\mathbb{F}$ ) has a root in $\mathbb{F}$ .

Interpretation¶

A field is algebraically closed when polynomial equations never force us to leave it: every equation $p(x)=0$ with $p$ a nonconstant polynomial with coefficients in $\mathbb{F}$ already has a solution in $\mathbb{F}$ .

The definition asks for one root, but this gives all of them. If $p(\lambda)=0$ $0$ , then polynomial long division by $x-\lambda$ leaves remainder $0$ , so $p(x)=(x-\lambda)q(x)$ with $\deg q=\deg p-1$ $1$ . If $q$ is nonconstant, it has a root as well, and repeating the step splits $p$ completely. So over an algebraically closed field, a polynomial of degree $n\ge1$ is a constant times a product of $n$ linear factors $x-\lambda_1,\ldots,x-\lambda_n$ $λ_{n}$ , where some $\lambda_i$ may coincide.

The word closed does not refer to the field operations; every field is closed under addition and multiplication. It refers to solving equations: in $\mathbb{R}$ the equation $x^2+1=0$ has no solution, and in $\mathbb{Q}$ the equation $x^2-2=0$ has none either. Enlarging $\mathbb{R}$ by $\mathrm{i}$ is enough to repair this for every polynomial at once, which is the content of the fundamental theorem of algebra: the complex numbers form an algebraically closed field.

This is one reason why complex numbers appear in linear algebra. Later, the eigenvalues of a square matrix turn out to be roots of a polynomial, so over $\mathbb{C}$ they always exist, even when the matrix has only real entries.

Key Results¶

The complex numbers $\mathbb{C}$ are algebraically closed (see the fundamental theorem of algebra).
The real numbers $\mathbb{R}$ are not algebraically closed.

Examples¶

$x^2+1$ has no root in $\mathbb{R}$ , but has roots $\mathrm{i}$ and $-\mathrm{i}$ in $\mathbb{C}$ .
$\mathbb{Q}$ is not algebraically closed: $x^2-2$ has no rational root.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 1

Aliases: algebraically closed Symbols: complex numbers, lambda, rational numbers, real numbers

Last updated: 20 October 2026 03:02