that should be not anyway, as i showed in this thread, the compositum of is so we only need to show that :

let and be an -basis for and respectively. then is an -basis for we define the map by

the fact that is a well-defined algebra homomorphism is a trivial result of the universal property of tensor product. it's also obvious that

is surjective. so we only need to prove that is injective. to prove this first show that separability of implies that is an -linearly independent set, for any integer

now suppose that which means that for any let then since is purely inseparable, there

exists an integer such that for all but then thus because, as i asked you to prove, is an

-linearly independent set. clearly implies that and hence for all because the set is an -basis for finally for all gives us