# Subextensions of Separable Field Extension are Separable

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## Theorem

Let $E / K / F$ be a tower of fields.

Let $E / F$ be separable.

Then $E / K$ and $K / F$ are separable.

## Proof

### Upper extension

We prove that $E / K$ is separable.

Let $\alpha \in E$.

Let $f$ be its minimal polynomial over $F$.

Let $g$ be its minimal polynomial over $K$.

Then by hypothesis, $f$ is separable.

On the other hand:

- $f \in K \sqbrk x$

and:

- $\map f \alpha = 0$

Hence by definition $g$ divides $f$.

By Divisor of Separable Polynomial is Separable, $g$ is separable.

$\Box$

### Lower extension

It follows immediately by definition that $K / F$ is a separable extension.

$\blacksquare$

## Also see

- Transitivity of Separable Field Extensions, the converse