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Let $K=M(X,Y)$ be the function field in two variables, for a field $M$ of characteristic $p>0$. Consider the Galois homomorphism $\gamma: f(X,Y)\mapsto f(Y,X)$ and let $F=K^\gamma$ the fixed field. Let $E=M(X^{1/p},Y)$, then $E/F$ is an extension of degree $2p$. By characteristic $p$, every element $b\in E$ satisfies $b^p\in K$ and therefore the extension $E/K$ is purely inseparable. Hence $E_\mathrm{sep}=K$. The element $\alpha=X^{1/p}Y$ of $E$ has minimal polynomial $g(Z)=Z^p-XY^p$ over $E_\mathrm{sep}$ and therefore $f(Z)=(Z^p-XY^p)(Z^p-X^pY)$ over $F$. Then $f$ has zeros $X^{1/p}Y$ and $XY^{1/p}$ the latter of which does not lie in $E$, therefore $E/F$ is not normal.