Separable Elements Form Field

It has been suggested that this page be renamed. In particular: Relative Separable Closure is Intermediate Field To discuss this page in more detail, feel free to use the talk page.

Theorem

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

Let $K$ be the relative separable closure of $F$ in $E$.

Then $K$ is an intermediate field of $E / F$.

Proof

We need to show that $K$ is a field.

By Transitivity of Separable Field Extensions, an algebraic extension generated by a family of separable elements is separable.

This theorem requires a proof. In particular: proof of Theorem 4.5 in 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.

Source

• 1996: Patrick Morandi: Field and Galois Theory: Chapter $1$: $\S4$: Separable and Inseparable Extensions: Proposition $4.20$
• 2002: Serge Lang: Algebra (Revised 3rd ed.): Chapter $\text V$: $\S4$: Separable Extensions: Theorem $4.5$