Algebraically closed field

Definition

A field $K$ is algebraically closed if every polynomial $f \in K[x]$ with $\deg f \ge 1$ has a root in $K$.

Remarks

Equivalently, every nonzero polynomial over $K$ splits over $K$. Also equivalently, the irreducible polynomials over $K$ are exactly the polynomials of degree one. Another equivalent condition: $K$ has no algebraic extension other than $K$ itself. The field $\mathbb{C}$ is algebraically closed, by the fundamental theorem of algebra. The field $\mathbb{R}$ is not, because $x^2 + 1$ has no real root. No finite field is algebraically closed: for a finite field $K$, the polynomial $1 + \prod_{a \in K} (x - a)$ has no root in $K$.

Explanations

L1 human

In an algebraically closed field, every polynomial equation of degree at least one has a solution. You never need a bigger field to find roots.