Is there a notion of Galois extension for Z / p^2? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T05:27:51Z http://mathoverflow.net/feeds/question/20773 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20773/is-there-a-notion-of-galois-extension-for-z-p2 Is there a notion of Galois extension for Z / p^2? Tran Chieu Minh 2010-04-08T19:20:08Z 2010-04-09T07:26:00Z <p>The above title is in fact a special case of what I want to ask. </p> <p>Certainly we have a well defined notion of Galois extension for $\mathbb{Q}_p$. The intersections of these extensions to the ring of integer of the absolute algebraic closure of $\mathbb{Q}_p$ give us a notion of Galois extensions for $\mathbb{Z}_p$. ( I know that there is a notion of Galois extension for commutative rings, and I believe that it should give us this. Am I correct?)</p> <p>Let's go further. Let $A_K$ be the ring of integer in a finite Galois extension $K$ of $\mathbb{Q}_p$. Let $e$ be the ramification degree of $K$ over $\mathbb{Q}_p$. The injection of $\mathbb{Z}_p$ into $A_K$ will induce an injection of $\mathbb{Z} / p^n$ into $A_K / \mathfrak{p}^{en}$. In this picture, there seems to be some desire to say that $A_K / \mathfrak{p}^{en}$ is the correct notion Galois of extension of $\mathbb{Z} / p^n$. But there are problems; taking this notion of Galois extension, if $K$ is has ramification degree $e >1$, the corresponding extension $A_K /p^e$ is not a field (it is not even an integral domain).</p> <p>Question 1: Is there any notion of Galois extensions corresponding to what I desire?</p> <p>Question 2: Can a class field theory (i.e a nice description of absolute abelian Galois extension) of $\mathbb{Z}/p^n$ be developed in this context? Is there any relationship between this and the local class field theory of $\mathbb{Q}_p$ ( which is the same as that of $\mathbb{Z}_p$)?</p> http://mathoverflow.net/questions/20773/is-there-a-notion-of-galois-extension-for-z-p2/20778#20778 Answer by Robin Chapman for Is there a notion of Galois extension for Z / p^2? Robin Chapman 2010-04-08T20:03:19Z 2010-04-08T20:03:19Z <p>There's the notion of Galois ring. Let $K$ be the degree $m$ unramified extension of $\mathbb{Q}_p$ and let $\mathcal{O}_K$ be its ring of integers. Then the quotient $R=\mathcal{O}_K/p^n\mathcal{O}_K$ is called the <em>Galois ring</em> of characteristic $p^m$ and residue field $\mathbf{F}_{p^m}$. </p> <p>The Frobenius map of $\mathbb{F}_{p^m}$ lifts to an automorphism of $R$ whose fixed ring is $\mathbb{Z}/p^n\mathbb{Z}$. There is a whole book on the topic: <a href="http://www.worldscibooks.com/mathematics/5350.html" rel="nofollow">http://www.worldscibooks.com/mathematics/5350.html</a> .</p> http://mathoverflow.net/questions/20773/is-there-a-notion-of-galois-extension-for-z-p2/20783#20783 Answer by dke for Is there a notion of Galois extension for Z / p^2? dke 2010-04-08T21:13:39Z 2010-04-08T21:13:39Z <p>Perhaps not directly answering your questions but something along those lines is Deligne's theory of truncated valuation rings, given in <a href="http://www.ams.org/mathscinet-getitem?mr=771673" rel="nofollow">Les corps locaux de caractéristique $p$, limites de corps locaux de caractéristique 0</a>.</p> <p>A truncated valuation ring is an Artin local ring with principal maximal ideal and finite residue field - by Cohen's structure theorem, these are precisely the quotients of rings of integers in local fields by a power of the maximal ideal. Deligne sets up a category using these truncated valuation rings and provides definitions of extensions aswell as a ramification theory for them. </p> <p>He goes on to show an equivalence between the category of "at most $e$-ramified" separable extensions of a local field $K$ and the category of "at most $e$-ramified" extensions of the length $e$ truncation of the ring of integers of $K$.</p> <p>The main point of all this is that the behaviour of objects defined over discrete valuation rings is often determined by their reduction modulo a power of the maximal ideal i.e. on truncated data. It also ties in with Krasner's idea (hence the title of Deligne's paper) that local fields of characteristic $p$ are limits of local fields of characteristic 0 as the absolute ramification index tends to infinity.</p>