Exercise 7.1.4

Proof.

(a)

Let $F\subset L$ a finite extension, where $F$ has characteristic $p$, and let

$K=\{\alpha \in L|\alpha$ is separable over $F\}$.

By Theorem 7.1.6 and Exercise 3 (noting that $1$, root of $x-1$ is in $K$), $K$ is a subfield of $L$. Moreover, every $\alpha \in F$ is root of the irreducible separable polynomial $x-\alpha \in F[x]$, so $\alpha$ is separable , thus $F\subset K$, and $F\subset K$ is a separable extension.

(b)

We show that the extension $K\subset L$ is purely inseparable.

Let $\beta \in L\setminus K$.

If $\beta$ was separable over $K$, then by Theorem 7.1.6, $K\subset K(\beta )$ would be a separable extension. But $F\subset K$ is also separable, thus by Theorem 5.3.15(c), $F\subset K(\beta )$ would be separable, and then $\beta$ would be separable over $F$, that is $\beta \in K$: this is a contradiction. No $\beta \in L\setminus K$ is separable over $K$, so the extension $K\subset L$ is purely inseparable.