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

Let $K = F^{sep}$ be the set of elements of $E$ which are separable over $F$.

This is a subextension by Separable Elements Form Field.

Then elements of $E \setminus K$ are not separable over $F$, since all elements that are separable over $F$ are in $K$.

Then elements of $E \setminus K$ are not separable over $K$, since if an element of $E$ is separable over $K$, then it is separable over $F$, by Transitivity of Separable Field Extensions.

By definition of purely inseparable field extension, $E/K$ is purely inseparable.

$\blacksquare$

Source

• 2002: Serge Lang: Algebra (Revised 3rd ed.): Chapter $\text V$: $\S 6$: Inseparable Extensions: Proposition $6.6$