My textbook says the extension $\mathbb{F}_p(T)/\mathbb{F}_p(T^p)$ is normal but not separable because the minimal polynomial of $T$ is not separable.
I know that the minimal polynomial of $T$ is $X^p-T^p$ but why isn't it separable? and why is this extension normal?
(1) The extension is normal because it is the splitting field of the irreducible polynomial $\;x^p-T^p\in\Bbb F_p(T^p)[x]\;$
(2) It isn't a separable polynomial since its unique root $\;T\;$ is multiple.