Please consider the following question:
Let F be an algebraic extension field of K , S the set of all elements of F which are separable over K, and P the set of all elements of F which are purely inseparable over K .Let E be an intermediate field. Then , if $E \cap S = K $ then show that $E\subset P$.
using the given information I can see that $ K \subset S$ and $ K \subset E$ but I am unable to see how P will co relate to E and unable to do so which is necessary.
SO, Can you please help me with this?
The statement is trivial in characteristic zero so we may assume that $\operatorname{char} K=p$. Take any $a\in E$. then we can write the minimal polynomial of $a$ over $K$ as $g(x^{p^r})$ where $g\in K[x]$ is irreducible and separable. Hence $a^{p^r}$ is a zero of $g$ and therefore separable. But by assumption $E\cap S=K$, so we get $a^{p^r}\in K$. Therefore $a$ is purely inseparable.
Note that this proof actually shows a bit more, namely that if $E\setminus K$ contains only inseparable elements, then it is already a purely inseparable extension.