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