If $K\subseteq L$ is a field extension and $x\in L$ is algebraic, we say that $x$ is separable over $K$ iff its minimal polynomial $f$ over $K$ is separable (i.e., $f$ is relatively prime with its derivative). We say that $L$ is a separable algebraic extension iff every element of $L$ is separable algebraic.
These definitions, of course, are quite standard. Now what I'd like to find is a proof of such standard facts as "if $x$ is algebraic separable over $K$ then $K(x)$ is separable" and "if $L$ is algebraic separable over $K$ and $M$ is such over $L$ then $M$ is such over $K$", or a definition of the separable degree of an extension, all without using field automorphisms or the trick of counting embeddings in an algebraic closure.
(There can be a number of reasons to want this: for pedagogical purposesout of a desire to postpone a discussion of Galois theory at a later point, or because embeddings/automorphisms are computationally or logically more complex objects than field extensions, or simply because it seems that the point of view in the first paragraph above should be more natural, or to compare different points of view.)
Now every textbook I could find on field extensions uses at some point a comparison between the number of embeddings of $L$ in the algebraic closure of $K$ and the degree $[L:K]$. But surely this can be avoided (we can, instead, work explicitly with roots of polynomials and perhaps elementary symmetric functions).
So, does someone know a place where separable field extensions are introduced without counting embeddings or similar objects, staying as close as possible to the definition I gave above?
Edit: Maybe the nicest definition of an algebraic $x$ being separable over $K$ of characteristic $p$ is that $K(x) = K(x^p)$.