Examples of extensions of a perfect field which are not separably generated Let $K$ be an extension field of a field $k$.
We say $K$ is separably generated over $k$ if $K$ has a transcendence basis $S$ over $k$ such that $K$ is separably algebraic over $k(S)$.
Let $k$ be a perfect field of characteristic $p \neq 0$.
Let $K$ be an extension field of $k$.
It is well-known that if $K$ is finitely generated over $k$, $K$ is separably generated over $k$.
I would like to know examples of $K$ which is not separably generated over $k$ in the following cases if any.
(1) tr.dim $K/k < \infty$
(2) tr.dim $K/k = \infty$
 A: Edit New proof. 
Let $k$ be a perfect field of characteristic $p>0$. Let $I$ be a non-empty set. Fix an algebraic closure $L$ of $k(t_i)_{i\in I}$  and consider 
$$K=k(t_i^{1/p^{\infty}})_{i\in I}:=\cup_{n\ge 1} k(t_i^{1/p^n})_{i\in I}\subseteq L.$$
We will show that $K$ can't be algebraic separable over a purely transcendental subextension $k(S)$. 
Suppose the contrary. The transcendental basis $S$ can be indexed by $I$ and we write $S=\{ s_i\}_{i\in I}$. Fix an index $i_0\in I$. Then $s_{i_0}$ is a fraction in some $t_j$'s. So $s_{i_0}^{1/p}$ exists in $K$ (here we use $k$ is perfect, but this doesn't really matter). But $K$ is separable over $k(S)$. Thus $s_{i_0}^{1/p}$ belongs to the purely trancendental $k(S)=k(s_i)_{i\in I}$. But this is obviously false.
Let $k'/k$ be any field extension. As $K$ is union of finitely generated subextensions $K_i$ and $K\otimes_k k'$ is the union (colimit to be exact) of the reduced $k$-algebras $K_i\otimes_k k'$, $K\otimes_k k'$ itself is also reduced. 
So any field extension $K/k$ over a perfect field $k$ is separable. 
