[joppy]

A field is algebraically closed if whenever you have a polynomial with coefficients in that field, all the roots of the polynomial also belong to the field.

For example: The field of rational numbers is not closed since it does not contain the roots of x^3 - 3 (since we can prove the cube root of 3 is irrational). The real numbers are not closed since they do not contain the roots of x^2 + 1. The complex numbers however, are closed.

An algebraic closure of a field is an embedding of that field into a closed field. For example, the rational numbers into the complex numbers, or the real numbers into the complex numbers. The second case is special: while rational numbers have many possible closures, the only closure of the real numbers is the complex numbers.