# Definition:Algebraically Closed Field

## Definition

Let $K$ be a field.

Then $K$ is **algebraically closed** if and only if:

### Definition 1

- The only algebraic field extension of $K$ is $K$ itself.

### Definition 2

- Every irreducible polynomial $f$ over $K$ has degree $1$.

### Definition 3

- Every polynomial $f$ over $K$ of strictly positive degree has a root in $K$.