Algebraic closure

Definition

Let $K$ be a field. An algebraic closure of $K$ is a field extension $\overline{K}/K$ such that $\overline{K}/K$ is algebraic and $\overline{K}$ is algebraically closed.

Remarks

The field $\mathbb{C}$ is an algebraic closure of $\mathbb{R}$, by the fundamental theorem of algebra. The field $\overline{\mathbb{Q}}$ of algebraic numbers is an algebraic closure of $\mathbb{Q}$. The field $\mathbb{C}$ is not, because $\mathbb{C}/\mathbb{Q}$ is not algebraic. If $K$ is infinite, then $\overline{K}$ has the same cardinality as $K$. If $K$ is finite, then $\overline{K}$ is countably infinite. An algebraic closure is a largest algebraic extension of $K$, and a smallest algebraically closed field that contains $K$.

Explanations

L1 human

The algebraic closure adds to a field all the roots of all its polynomials, and nothing else. For the rationals, it gives the algebraic numbers, not all the complex numbers.