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 $K$.

Also see

• Equivalence of Definitions of Algebraically Closed Field
• Definition:Separably Closed Field