On a formula of the norm of an element of a finite extension of a field Theorem
Let $F$ be a field.
Let $K$ be a finite extension of $F$.
Let $[K : F]_i$ be the inseparable degree of $K/F$.
Let $\bar{K}$ be an algebraic closure of $K$.
Let $S$ be the set of $F$-embeddings of $K$ into $\bar{K}$.
Let $\alpha \in K$.
Then $N_{K/F}(\alpha) = (\prod_{\sigma \in S}\sigma(\alpha))^{[K : F]_i}$
This is proved in Theorem 60 in page 39 of the lecture note written by Pete L. Clark.
I don't understand the proof.
Would any one please enlighten me?
EDIT
Why down votes??
Asking for help to understand a proof should be frowned upon?
EDIT
I understand the prerequisites for the proof, i.e. the content of section 6 and Corollary 58.
EDIT
Related question 1,
Related question 2
EDIT
Let $f(X)$ be the characteristic polynomial of $\alpha$.
Let $g(X)$ be the minimal polynomial of $\alpha$.
By Corollary 58, $f(X)$ = $g(X)^{[K:F(\alpha)]}$.
The set {$\sigma(\alpha); \sigma ∈ S$} is the set of the roots of g(X).
However, it is not clear to me that the equation $N_{K/F}(\alpha) = (\prod_{\sigma \in S}\sigma(\alpha))^{[K : F]_i}$ follows immediately.
Would anyone please explain why the equation follows other than just saying it is straightforward?
EDIT
It's amazing that some people regard questioning a proof as an attack to the author's credibilty. Everybody makes a mistake. Even Grothendieck made a non-trivial mistake(see Misconceptions About $K_{X}$ by Kleiman).
I think this attitude is harmful to healthy development of mathematics.
I'm not claiming that the proof is wrong, though.
EDIT
To anyone who thinks the proof is straightforward, please explain to me in detail.
I am not as smart as you.
EDIT
It's surprising that no one explained the proof so far.
I believe every step of any proof can be reduced to (really) trivial statements.
If you think it's straightforward, please reduce it to more trivial statements that anybody who has basic knowledge of abstract algebra can understand.
 A: I don't think the downvotes are helping the situation.  In case anyone is downvoting the OP's question out of some sense of solidarity with / loyalty to me: thanks, but please don't.  
On the other hand, exactly what the OP wants to ask is not clear, even to me.  "I don't understand the proof.  Would any one please enlighten me?" isn't specific enough to be helpful.  In fact, more than a couple of days ago now, this started with a comment the OP left to me elsewhere on this site:
"@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."
Let me start by saying that I most certainly do want errors in my mathematical writings to be brought to my attention.  At this point though I have about 2000 pages of such things available on the internet, so it is a nontrivial matter as to how to best deal with adding / editing / correcting these things.  Generally at any point in time I am more mentally present and interested in making these changes on some documents than others (if anyone cares, at the moment it is honors calculus / undergraduate analysis, geometry of numbers and quadratic forms which have most of my attention).  
What I'm trying to say is: at the moment I don't really have a well thought out system to deal with modifications to the product I've already made available.  People write me to suggest changes and corrections fairly frequently (a few times a month, on average), and I'm sorry to say that I address some but not all of these suggestions in a timely fashion.  For now, if you want to maximize the chances that a correction gets made sooner rather than later, then it helps to give me as much information as possible and give me as concrete a task as possible.  For instance:
1) Please include specific information about which notes you are looking at: a link to a particular file on my webpage is ideal.  
2) If you think that I've made a mistake, please say exactly what you think the mistake is.  If you think you know the correction, of course please tell me.  
3) If you just didn't understand something in my notes, or have a natural followup question, or are seeking references on something I alluded to too casually (N.B.: I have gotten much better in recent years at including precise, formal references even in lecture notes for my undergraduate classes), then you might want to try asking someone else first, or asking on this site.  When it comes to teaching, I will naturally put my own students, and students at my own university, first.  Anyway, when it comes to questions about my expository notes, I promise you that I am not uniquely or best qualified to answer them.  On the contrary, for almost anything that I have written about, there are people here who are more qualified to address your questions than I am.  
If you do ask a question here that comes from my own writings, please ping me.  I will be interested to read it, even if I am not one of the ones to answer it.
A: What exactly you don't understand in that proof? It is heavily based on former results in the same notes, and if you know already that the conjugates of certain element are the images of this element under the different embeddings $\,K\to \overline{K}\,$, each being repeated precisely $\,[K:F]_i\,$ , then...well, then we're done!
A: Well the theorem is a bit too hastily stated, for example if you happen to pick $\alpha \in F$, then $f(t) = (t-\alpha)^n$ and has only one distinct root.
The correct statement is :
Let $K/F$ be a field extension of degree $n < \infty$ and separable degree $m$. Let $\overline{K}$ be an algebraic closure of $K$. Let $\alpha \in K$ and let $f(t)$ be the characteristic polynomial of $\alpha \bullet \in End_F(K)$. Let $\sigma_1 \ldots \sigma_m$ be the distinct $F$-algebra embeddings of $K$ into $\overline{K}$. Then the factorisation of $f(t)$ on $\overline{K}$ is $f(t) = \prod_{i=1}^m (t- \sigma_i(\alpha))^{n/m}$.
First, let $K_0 = F(\alpha)$, $n_0 = [K_0 : F]$, $m_0$ be the separable degree of $[K_0 : F]$, $\tau_1 \ldots \tau_{m_0}$ be the embeddings of $K_0$ into $\overline{K}$, and $f_0(t)$ be the minimal polynomial of $\alpha$ over $F$.
Then, by corollary 58, $f(t) = f_0(t)^{[K : K_0]} = f_0(t)^{n/n_0}$.
According to the proof of theorem 38, each $\tau_i$ is the restriction to $K_0$ of $m/m_0$ many embeddings $\sigma_j$, therefore
$\prod_{i=1}^m (t- \sigma_i(\alpha))^{n/m} = \prod_{i=1}^{m_0} (t- \tau_i(\alpha))^{n/m_0} $, so we only need to prove that $f_0(t) = \prod_{i=1}^{m_0} (t- \tau_i(\alpha))^{n_0/m_0}$.
From now, suppose we are in the case $K = K(\alpha)$. Let $K_s$ be the separable closure of $F$ in $K$, so by Proposition 50, $K/K_s$ is purely inseparable and $K_s/F$ is separable. By corollary 47, there is an integer $a\ge 0$ such that $K_s = K(\alpha^{p^a})$.
Let $g(t)$ be the minimal polynomial of $\alpha^{p^a}$ over $F$.
Then $f(t) = g(t^{p^a})$, because $g(\alpha^{p^a}) = 0$ and they have the same degree $n = p^a m$.
Meanwhile, $(t - \sigma_i(\alpha))^{p^a} = (t^{p^a} - \sigma_i(\alpha^{p^a}))$ so we only need to prove that $g(t) = \prod_{i=1}^m (t - \sigma_i(\alpha^{p^a}))$.
So we only need to prove the theorem when $\alpha$ is separable.
But then this is clear : the $\sigma_i(\alpha)$ are roots of $f(t)$ because $\sigma_i$ are embeddings of $F$-algebras, and every factor in the product is distinct, so the product has to divide $f(t)$, and both polynomials have the same degree, so they must be equal.
