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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.

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Note that Lang sketches the proof that these two definitions give the same maps in Proposition IV.5.6. – Dylan Moreland Jul 11 '11 at 2:06
Also, I can explain why Lang's definition makes sense, but it isn't clear to me that this would be welcome. – Dylan Moreland Jul 11 '11 at 2:23
up vote 5 down vote accepted

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.

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Thanks for these notes, Pete. I am still struggling with one question I had at the beginning. What does it mean, when one says, "Let $\sigma_1,...,\sigma_n$ be the embeddings of $K$ into its algebraic closure". I might have to review some algebraic closure, I guess. – B M Jul 11 '11 at 2:26
@Brittany: sorry, for some reason I wasn't completely sure after reading your question that it was the embeddings into the algebraic closure part that you found satisfactory. (It looks clear enough now.) In the long run at least it would certainly be best for you to learn about this material: it's not very hard, and it comes up in a lot of places. – Pete L. Clark Jul 11 '11 at 2:53
In my text, this material gets covered in $\S 6.2$. (I notice that my personal copy has fixed up this section a little bit. These notes could be a lot better, I'm afraid.) But if you're willing to assume that your extension $K/F$ is separable, then it follows that it is of the form $K = F[t]/(P(t))$ for an irreducible polynomial $P$, so the embeddings of $K$ into any extension field $L$ of $F$ are obtained by sending the canonical root $t + (P(t))$ of $P$ in $K$ to any root of $P$ in $L$. So there are at most degree of $P$ = degree of $K$ $F$-embeddings of $K$ into any field... – Pete L. Clark Jul 11 '11 at 2:57
...and for any field over which any polynomial splits into linear factors there will be exactly that many embeddings. That should be most of what you need to know. – Pete L. Clark Jul 11 '11 at 2:58
@Pete I think the proof of Theorem 60 is incorrect(I'm not talking about some obvious typos there). Please forgive me if this is my mistake. – Makoto Kato May 8 '12 at 7:28

I have some notes on the matter online here, including the equivalence of the definitions. The proofs were based on my lecturer's notes.

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Thanks, Zhen. It will be good to read through the various references in case I get stuck with one. – B M Jul 11 '11 at 2:28

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.

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Thanks, Adrian. I have read Robert Ash's notes on commutative algebra, so this should be a good reference. – B M Jul 11 '11 at 2:27

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.

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