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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!

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    $\begingroup$ Have you tried Lang's Algebra? $\endgroup$
    – Corey
    Commented Jul 7, 2011 at 7:34
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    $\begingroup$ An exposition of the characterization of abelian $p$-extensions of a field of characteristic $p$ in terms of the ring of Witt vectors is given in chapter 8 of the ever useful Basic Algebra II by Jacobson. If you want just a proof for the fact that a cyclic extension of char $p$ fields is of the form $E=F(y)$, where $y^p-y=z$ for some $z\in F$, then I can write it out. But undoubtedly you've seen those (or done it as an exercise), and are looking for something more :-) $\endgroup$ Commented Jul 7, 2011 at 7:40
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    $\begingroup$ (i) It's ironic that "Artin-Schreier theory" could refer either to the theory of fields with finite nontrivial absolute Galois group (which are necessarily of characteristic zero) or to the theory of cyclic $p$-extensions in characteristic $p$. I think you are asking about the latter, but I'm not completely sure. (ii) AS theory in the former sense is to be found in my notes-in-progress on field theory. I keep looking there for AS theory in the latter sense: so far I've never found it in my lecture notes, to my disappointment. But some day... $\endgroup$ Commented Jul 7, 2011 at 7:59
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    $\begingroup$ P.S.: I'm having trouble imagining what embarrassing difficulties looking up Artin-Schreier theory might be. But anyway, I'm pretty sure it's not your fault: it's not as easy to find as it should be. Don't be embarrassed! $\endgroup$ Commented Jul 7, 2011 at 8:02
  • $\begingroup$ Thank you for your time! I'm actually looking for the theory of cyclic $p$-extensions in characteristic $p$. I will try to look to the books you suggested! In specific I was looking for the proof of the fact that a Galois extension (not merely cyclic, but maybe it is obvious how to extend the result) of $char(p)$ fields is of the form Jyrki suggested (as stated on the wiki page), but I have had troubles for an handmade proof. Thank you again! $\endgroup$ Commented Jul 7, 2011 at 8:55

1 Answer 1

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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 $m$.

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  • $\begingroup$ Thank you! I really appreciated this solution, now if I understand correctly this proof classifies all the abelian Galois extensions of degree $p$ (prime degree + abelian = cyclic). And if I want study extensions of degree a power of $p$ I need Witt vectors, am I right? I've still a doubt, what about non-abelian Galois extension? They does exist? How are they classified? Thank you again! Including the excellent reference you suggested the answer is exactly what I was looking for! $\endgroup$ Commented Jul 7, 2011 at 10:24
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    $\begingroup$ The theorem on linear independece of character should be dued to Artin. $\endgroup$ Commented Jul 7, 2011 at 10:32
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    $\begingroup$ By the way, note that $\sum \sigma^k(x)$ is just $Tr(x)$. So, instead of using independence of characters, one can quote the fact that the trace map is nonzero for a separable extension. (I'm not sure which of those is easier to prove.) $\endgroup$ Commented Jul 7, 2011 at 11:37
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    $\begingroup$ The Artin-Shreier-Witt theory describes the abelian p-extensions of a field of characteristic p. Unfortunately I don't know any book form account, but a good introduction for beginners is the 1rst chapter of Lara Thomas' thesis, " Arithmétique des extensions d'ASW ", Univ. Toulouse Le Mirail . $\endgroup$ Commented Apr 12, 2016 at 7:48
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    $\begingroup$ To supplement this nice answer: Lang's Algebra has a series of exercises which builds the theory of Witt vectors and deduces the Artin-Schreier-Witt theory of finite abelian extensions (at the end of the chapter of Galois theory). Serre's Local Fields, section X.3, gives a quick cohomological interpretation. $\endgroup$
    – Yai0Phah
    Commented Feb 10, 2020 at 10:14

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