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.