Self contained reference for norm and trace I have learnt about the concepts of norm and trace of an element with respect to a finite extension, say $L/K$, of fields in terms of the determinant and trace (resp.) of the corresponding scalar multiplication map. 
I looked these concepts up in Lang and his formulation is in terms of embeddings of the field $L$ into an algebraic closure of $K$. I don't quite understand this definition and it's equivalence with the above definition (specifically because the number of embeddings and their choice seems arbitrary). 
I am looking for a self contained reference of the above concepts as done in Lang. Of course, a trivial answer is Lang, but I am looking for a reference which develops only enough machinery in order to define these concepts and probably discuss some of their properties. I eventually plan to read Lang from the ground up, but I need to cover these concepts quickly for now. 
 A: This is done in $\S 7$ of my (far from finished) field theory notes.  There is really nothing fancy going on here (not as much as I might want, even): it should be readable without much background.  As I say, one of my sources was a text of G. Karpilovsky.
I should say though that this is one of the topics that seems to set apart graduate-level treatments of field theory from undergraduate-level treatments of field theory: you'll almost always find a decent treatment of traces and norms in the graduate-level texts, whereas it's pretty rare to find it in the undergraduate-level texts.  For instance, I am currently spending a very pleasant evening with my copy of Jacobson's Basic Algebra: he discusses this in Volume I, Chapter 7.
A: I have some notes on the matter online here, including the equivalence of the definitions. The proofs were based on my lecturer's notes.
A: You can find it in Robert Ash's notes on Algebraic Number Theory in his webpage. Look at chapter two on norms, traces and discriminants. The proposition that proves what you want is Proposition 2.1.6
He defines the norm and trace using the determinant and the trace of the scalar multiplication map and then proves that they are the same as the corresponding products and sums of the corresponding values of the different embeddings.
By the way, his notes are available in book form from Dover publications for just a few dollars.
A: As usual, there is a Keith Conrad handout, which gives a few example calculations and avoids using the word "separable". Brian Conrad's handout goes through the example of a general quadratic extension, mentions explicitly some issues involving inseparable extensions (the trace map is zero, but the norm may be nontrivial), and proves that the norm is transitive, which is apparently messy. They both start from your favored definition.
