Purely inseparable subextension of $F(x)$

Question

Suppose that $F$ is a field of characteristic $p>0$. Prove field extension $F(x^p)\subset F(x)$ is purely inseparable.

I think we should first prove that is inseparable, and then show it is purely inseparable.

Answer 1

Put $K=F(x^p)$, $L=F(x)$. $x$ is a root of the polynomial $X^p-x^p\in K[X]$. Since $F$ is of characteristic $p$, $X^p-x^p=(X-x)^p$, which implies $x\in L$ is a inseparable element and $L/K$ is inseparable. Now for any $y\in L=F(x)$, write $y=f(x)/g(x)$ for some polynomials $f,g$ with coefficients in $F$. $y^p=f(x)^p/g(x)^p=f(x^p)/g(x^p)\in K$, so $X^p-y^p\in K[X]$. Again $X^p-y^p=(X-y)^p$. Thus $y$ is a inseparable element and $L/K$ is purely inseparable.