# 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 $f$ is separable.

Then $f \in K[x]$ and $f(\alpha) = 0$, hence by definition $g$ divides $f$.

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

$\Box$

### Lower extension

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

$\blacksquare$

## Also see

- Transitivity of Separable Field Extensions, the converse