Can someone please give me an example of a field extension $F\subset E$ such that E\F has both separable and inseparable elements?
if $F(\alpha)$ is a simple extension of F, and if $\alpha$ is separable... are all elements of $F(\alpha)$ separable?
thanks
As mentioned in the comments, for a field extension $F \subset E$ which contains separable as well as inseparable elements, simply adjoin such elements.
For an explicit example, let $p$ be a prime and $F=\mathbb F_p(t)$ the field of rational functions over the finite field $\mathbb F_p$. (You need an infinite field of positive characteristic to find inseparable extensions.) If $q$ is any prime then the polynomial $f_q = x^q - t \in \mathbb F_p(t)[x]$ is irreducible by an application of Gauss' Lemma and the Eisenstein criterion. Looking at $f_q'$, the polynomial $f_q$ is separable if $q\ne p$ and inseparable if $p=q$. So let $E =F(\alpha,\beta)$ where $\alpha$ is a root of $f_p$ and $\beta$ is a root of some $f_q$ for $q \ne p$.
A field extension $F \subset E$ is called separable if every element of $E$ is separable over $F$. As you suspected, one has the following.
Theorem Let $F\subset E$ be a field extension, such that $E=F(\alpha_1,\ldots,\alpha_n)$ with all $\alpha_i$ separable over $F$. Then the extension $F \subset E$ is separable.
To prove this, one characterizes separability by means of the separability degree, and uses a result about the extension of homomorphisms to simple extensions. Most textbooks covering separable field extensions should contain a proof of this. See, for instance, Lang, Algebra, Chapter V, Theorem 4.4.