Transitivity of Separable Field Extensions

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Theorem

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

Let $E / K$ and $K / F$ be separable.

Then $E / F$ is separable.

Proof

This theorem requires a proof. In particular: Using Chapter V, $\S4$, Corollary 4.2 of Lang's Algebra You 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.

Also see

• Subextension of Separable Field Extension is Separable, the converse

Source

• 1996: Patrick Morandi: Field and Galois Theory: Chapter $1$: $\S 4$: Separable and Inseparable Extensions: Proposition $4.21$
• 2002: Serge Lang: Algebra (Revised 3rd ed.): Chapter $\text V$: $\S4$: Separable Extensions: Theorem $4.3$