# Definition:Separably Closed Field

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## Definition

Let $K$ be a field.

Then $K$ is **separably closed** if and only if

### Definition 1

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

### Definition 2

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

### Definition 3

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

## Also see

- Equivalence of Definitions of Separably Closed Field
- Definition:Perfect Field, a field with no inseparable extensions
- Definition:Algebraically Closed Field