At+
César

1. The original phrasing is correct – g(x) is not irreducible. We are using it in order to create the polynomial f(x) which will be irreducible.
2. You are right, thanks!
3. Right.
4. v*u = sig-1(sig(u)*sig(v)) = sig-1((p+f(x))*(s+f(x))) = sig-1(p*s+f(x)) = sig-1(1+f(x)) = 1.
5. As the article says – the Galois group is the automorphism group of E/F. Part 2 of the article will explain the importance of the Galois group.

2.- I think that the symbols “” are swapped on the
first line of Gauss´s Lemma proof
“()”
That is:
(=>) If f(x) splits on Z, obviously it splits on Q, since Z is contained on Q.

3.- Whitout loss of generality I must assume that
“WLOG”
stands for
“Whitout Loss Of Generality”,
right?

4.- I can cope with the arguments up to page 3 where is claimed that
“Let v = \sigma^{-1}(s). It is clear that v \times u = 1″
justifying the claim with
“(if this is not clear to you at this point try to prove it)”
well I recognize that I tried to prove it and was unable to do it, so:
can you sketch the proof giving working hints to a standard proof?

5.- About the “Fields Automorphisms” sections: elsewhere I found a
figure relating what seemed to me as extension fields (perhaps splitting fields)
and Galois Groups, but there was not any explication and I can not
reformulate the underline relation myself. So, it would be great if you
can explain that relation.

Regards
César

The topic is covered in the (excellent) Algebra B 2 course.

For the second part of your comment: most of the algebra books I read did postpone the proof of the existance of the said polynomial until the end. Actually, as is mentioned in the introduction of the article I chose to present many thing in the reverse order. Now, I completely agree with you that the existance of such polynomials is not as important to the subject as the fundamental theorem, for example, but I really liked the proof!

A little criticism: The proof that you provided in part 3, as well as the related lemmas, are quite elegant. However, I feel that it is much easier for the reader to ‘believe’ in the existence of such a polynomial without proof than in the theorem about it. Therefore, unless it was necessary for developing the rest of the theory, I think that part 3 might have been better fit in a later installment. Nonetheless, it made for an enjoyable read by itself.