In mathematics, a field F is algebraically closed if every non-constant polynomial with coefficients in F has a root in F. In other words, a field is algebraically closed if the fundamental theorem of algebra holds for it. For example, the field of real numbers is not algebraically closed because the polynomial ${\displaystyle x^{2}+1}$ has no real roots, while the field of complex numbers is algebraically closed.

Every field ${\displaystyle K}$ is contained in an algebraically closed field ${\displaystyle C,}$ and the roots in ${\displaystyle C}$ of the polynomials with coefficients in ${\displaystyle K}$ form an algebraically closed field called an algebraic closure of ${\displaystyle K.}$ Given two algebraic closures of ${\displaystyle K}$ there are isomorphisms between them that fix the elements of ${\displaystyle K.}$

Algebraically closed fields appear in the following chain of class inclusions:

As an example, the field of real numbers is not algebraically closed, because the polynomial equation ${\displaystyle x^{2}+1=0}$ has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraically closed field is the field of (complex) algebraic numbers.

No finite field F is algebraically closed, because if a₁, a₂, ..., a_n are the elements of F, then the polynomial (x − a₁)(x − a₂) ⋯ (x − a_n) + 1 has no zero in F. However, the union of all finite fields of a fixed characteristic p (p prime) is an algebraically closed field, which is, in fact, the algebraic closure of the field ${\displaystyle \mathbb {F} _{p}}$ with p elements.

The field ${\displaystyle \mathbb {C} (x)}$ of rational functions with complex coefficients is not closed; for example, the polynomial ${\displaystyle y^{2}-x}$ has roots ${\displaystyle \pm {\sqrt {x}}}$, which are not elements of ${\displaystyle \mathbb {C} (x)}$.

Given a field F, the assertion "F is algebraically closed" is equivalent to other assertions:

The field F is algebraically closed if and only if the only irreducible polynomials in the polynomial ring F[x] are those of degree one.