I'm reading Janos Kolloar's Lecture on Resolution of Singularities and have an question about an argument used in the proof of Theorem 1.33 (page 25):
THEOREM 1.33 Let $S$ be an integral domain that is finitely generated over a field $k$, and let $F \subset Frac(S)$ be a finite field extensionof it's quotient field. Then the normalization of $S$ in $F$ is finite over $S$.
the proof strategy is: By Noether normalization theorem $S$ is finite over $R=k[x_1,...,x_n]$ thus it suffuce to consider the finite field extension $k(x_1,...,x_n) \subset F$.
Proof. [...] $F$ is a finite extension of $k(x_1,...,x_n)$ so there is a finite purely inseparable extension $E/k(x_1,...,x_n)$ such that $EF/E$ is separable. (???) Every finite purely inseparable extension of $k(x_1,...,x_n)$ is contained in a field $k'(x_1^{p^{-m}},...,x_n^{p^{-m}})$, where $k'/k$ is finite and purely inseparable (???) [...]
Questions:
Q_1: Why the author talks explicitly about finite purely inseparable extension $E/k(x_1,...,x_n)$ such that $EF/E$ is separable? Why he not simply says that $F$ contains an intermediate field $E$ such that extension $E/k(x_1,...,x_n)$ is finite purely inseparable and $F/E$ is separable? Or is this in general wrong claim? That is why we need the field $EF$ in the game?
Q_2: Why every finite purely inseparable extension of $k(x_1,...,x_n)$ is contained in a field $k'(x_1^{p^{-m}},...,x_n^{p^{-m}})$, where $k'/k$ is finite and purely inseparable?