The real numbers are not, however, an algebraically closed field.

Because of this fact, C is called an algebraically closed field.

Let C be an algebraically closed field and x a variable.

Here more generally one can consider algebraically closed fields of prime characteristic.

Over any algebraically closed field, there is just one Albert algebra.

Any extension of an algebraically closed field is regular.

Over an algebraically closed field it is equivalent to diagonalizable.

(This is clear from the above discussion for algebraically closed fields.

Let G now be a connected reductive group over an algebraically closed field.

These conditions are equivalent over an algebraically closed field of characteristic zero, such as the complex numbers.

Not all matrices are diagonalizable (even over an algebraically closed field).

It is a complete and algebraically closed field.

Now consider the class of algebraically closed fields.

The absolute Galois group of an algebraically closed field is trivial.

Algebraically closed fields with a (Krull) valuation are perhaps the most important example.

Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.

This includes complete theories of vector spaces or algebraically closed fields.

The classic example is the theory of algebraically closed fields of a given characteristic.

Any linear algebraic group over an algebraically closed field is totally transcendental.

See, for example, algebraically closed field or compactification.

Over an algebraically closed field, all Cartan subalgebras are conjugate.

It holds more generally for minimal projective surfaces of general type over an algebraically closed field.

The theory of algebraically closed fields has quantifier elimination.

The C fields are precisely the algebraically closed fields.

In general, algebraically closed fields are easier to handle than non-algebraically closed ones.