# Definition:Separable Polynomial

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## Contents

## Definition

Let $K$ be a field.

Let $\map P X \in K \sqbrk X$ be a polynomial of degree $n$.

### Definition 1

$P$ is **separable** if and only if its roots are distinct in an algebraic closure of $K$.

### Definition 2

$P$ is **separable** if and only if it has no double roots in every field extension of $K$.

### Definition 3

$P$ is **separable** if and only if it has $n$ distinct roots in every field extension where $P$ splits.

## Also defined as

A **separable polynomial** is also seen defined as a polynomial whose irreducible factors are separable in the sense of the definition above.