Reference book for Artin-Schreier Theory

The aim of the question is very simple, I would like to study Artin-Schreier Theory, but I have had embarassing difficulties in finding a book which could help me in doing that.

In specific I'm looking for a book which write down explicitely the main theorem of Artin-Schreir Theory, and have also a proof of it (or at least a sketch of it) and maybe also a short introduction on the topic.

The closest I get in the matter is with the book "Cohomology of Number Fields" of Neukirch, which is suggested as a reference by the Wikipedia page on Artin-Schreier Theory. But it doesn't fit with the requests I made above.

So the question is, what is the name of such a book? Or, if such abook does not exist, do you have any suggestions on how to proceed?

Thank you very much!

Solution 1:

Let $L/K$ be a cyclic Galois extension of order $p= char K$. Let $\sigma$ be a generator of the Galois group. By the 'independence of characters' theorem (don't remember for sure, whether it is due to Dedekind, Kummer or even Artin), there exists an element $x\in L^*$ such that $z=x+\sigma(x)+\sigma^2(x)+\cdots+\sigma^{p-1}(x)\neq0.$ Let us fix such an element $x$. Note that $z\in K$, because it is invariant under $\sigma$.

Write $$ y=(p-1)x+(p-2)\sigma(x)+\cdots+2\sigma^{p-3}(x)+\sigma^{p-2}(x)+0\cdot\sigma^{p-1}(x). $$ We see that $\sigma(y)=y+z$, so if we denote $u=y/z$, we get $\sigma(u)=u+1$. Therefore $u\notin K$, so $L=K(u)$. Fermat's little theorem tells us that $p(T)=T^p-T=\prod_{i=0}^{p-1}(T-i)$ in $K[T]$. In characteristic $p$ we have $p(a+b)=p(a)+p(b)$ for all $a,b\in L$. The minimal polynomial of $u$ is thus $$ \prod_{i=0}^{p-1}(T-\sigma^i(u))=\prod_{i=0}^{p-1}(T-(u+i))=p(T-u)=p(T)-p(u)= T^p-T+(-1)^p\prod_{i=0}^{p-1}(u+i), $$ so $L/K$ is of the AS form.

This is an additive analogue of the standard multiplicative argument (=starting point of Kummer theory) telling us that a cyclic extension of degree $m$ is a root extension, when the smaller field has a primitive root of unity of order $m$.