# Decomposition of Field Extension as Separable Extension followed by Purely Inseparable

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## Theorem

Let $E/F$ be an algebraic field extension.

Then the relative separable closure $K=F^{sep}$ in $E$ is the unique intermediate field with the following properties:

- $K/F$ is separable.
- $E/K$ is purely inseparable.

## Proof

This theorem requires a proof.In particular: use transitivity of separable extensionsYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |