So far I’ve only discussed finite Galois theory, that is the Galois theory of finite extensions. But there’s also an infinite theory; the extensions it studies are separable and normal, just like finite Galois extensions, but the fundamental theorem needs to be modified to apply. I could transfer it almost in its entirety by using the theory of topological groups, but I think it’s simpler to just make exceptions and note the equivalent topological terminology where necessary.
Infinite Galois theory is wedded to the theory of projective limits of groups, which is based on directed systems.
A set I is a directed system if it comes equipped with a partial order, which I’ll denote <–, such that given any two elements i and j of I, there exists a k in I such that i <– k and j <– k. For example, a total order induces a directed system. For a less trivial example, let I = N (without zero), with a <– b iff a divides b; given any i and j, a suitable k is their least common multiple. Note that if I has a maximal element, then that maximal element is unique.
Now, an inverse system of groups is a family of groups G(i) indexed by the set I, and homomorphisms from G(j) to G(i) whenever i <– j, denoted f(ij). The homomorphisms must be consistent with one another, i.e. given i <– j <– k, every g in G(k) must satisfy . Typically the homomorphisms are taken to be surjective, but strictly speaking this is not necessary.
For example, if I = N with the normal ordering, letting G(n) = Z/(p^n)Z and f(mn) be the obvious remainder map from Z/(p^n)Z to Z/(p^m)Z yields an inverse system. Alternatively, if I has the divisibility ordering defined above, then G(n) = Z/nZ with the remainder maps is similarly an inverse system.
An inverse system has a limit, defined to be the group that projects on all groups in the system. More precisely, let P be the product of all groups G(i). Note that this is a product rather than a direct sum, the difference being that in a direct sum all but finitely many terms must be the identity element. The inverse or projective limit is the subgroup of P consisting of all elements g in P such that for all i <– j in I, .
Finally living up to my blog’s name, I’ll also prove the abstract nonsense definition of projective limits. The projective limit comes with obvious projection maps ; if each f(ij) is surjective, then so is each f(i). Furthermore, taken together with those projection maps, it’s universal with the property that . In plain mathematical English, it means that if H is another group with projection maps h(i) such that f(ij)h(j) = h(i), then there exists a unique group homomorphism p from H to the projective limit such that h(i) = f(i)p.
This is likely the first time you encounter universal properties – if it is, consider yourself lucky to have never seen these monsters until now – so I’ll prove it in full instead of appeal to other universal properties. The idea is to explicitly define p in such a way that will yield h(i) = f(i)p.
Every a in H induces an element for each i in I. Furthermore, the condition that f(ij)h(j) = h(i) implies that . Therefore, the element of P whose ith entry is is in fact in the projective limit, yielding a function p from H to . This is a group homomorphism, roughly because the ith element of p(ab) is . Further, that the ith element of p(a) is implies immediately that h(i) = f(i)p.
Conversely, this homomorphism is unique because the condition that h(i) = f(i)p forces the ith element of p(a) to be . This is important because it shows that if H and H’ satisfy the universal property of the projective limit of the same inverse system of groups, then there exists a unique p with h(i) = h‘(i)p and a unique p‘ with h(i)p = h‘(i) so that p and p‘ are isomorphisms that are inverses of each other, and the universal property uniquely defines the projective limit.
Note that it’s possible to define similar inverse limits of every algebraic structure – rings, modules, algebras, and so on – as well as of topological spaces. Also note that if I has a maximal element then the projective limit is just that element, making the case of interest the one with no maximal element. Finally, duplicating an index – i.e. splitting i into i1 and i2, with i(j) <– k iff i <– k and k <– i(j) iff k <– i, and with i1 and i2 either comparable or incomparable – doesn’t change the projective limit.
As an example, the projective limit of G(n) = Z/(p^n)Z is , the additive group of the padic integers; it’s certainly true that every sequence of residue classes modulo p^n compatible under taking remainders induces a padic integer and vice versa.
More interestingly, the projective limit of G(n) = Z/nZ is the product of Z(p) over all primes p. This is because by the universal property, , and Z/(p1^a1)(p2^a2)…(p(k)^a(k))Z = Z/(p1^a1)Z * Z/(p2^a2)Z * … * Z/(p(k)^a(k))Z.
Back to Galois theory now. If L/K is an infinite Galois extension (i.e. separable and normal, as in the finite case), then the Galois groups of the intermediate Galois extensions form an inverse system, roughly because if F/E/K is a tower of Galois extensions, then Gal(F/K) projects onto Gal(E/K). If you deal with sequences better than with abstract inverse systems, then let L be the splitting field of {f1, f2, …} and look at the splitting field of f1, then this of f1f2, then this of f1f2f3…
A Kautomorphism of L is defined by its action on each a in L. But L/K is assumed to be algebraic, so each a in L is contained in a finite extension of K, which has a Galois closure, say F. As F/K is normal, every Kautomorphism of any extension of F will map F to itself; this was proved together with the part of the fundamental theorem concerning normal subgroups of the Galois group and normal extensions. Therefore, a Kautomorphism of L is completely determined by the automorphisms it induces on each finite Galois extension F/K.
Finally, to see that Gal(L/K) is the inverse limit of Gal(F/K) over all intermediate F, note that given a tower of Galois extensions L/F/E/K, the map from Gal(L/K) to Gal(E/K) is fairly obviously the same as the map from Gal(L/K) to Gal(F/K) composed with the one from Gal(F/K) to Gal(E/K).
Here is where the standard Galois correspondence breaks down. To see why, let K = F(p) for any p, and L be the union of F(p^(q^n)) over all n; then Gal(L/K) is Z(q). The group Z(q) is uncountable and so has uncountably many subgroups, but L/K only has one proper intermediate extension for each integer n.
It’s necessary to view the projective limit as not just a group but also the inverse system it’s based on and the homomorphisms to each of the groups in the system. In this view, a subgroup must come from the groups in the system. More precisely, a subprojective limit of G(i) arises as the projective limit of a collection of subgroups H(i) such that f(ij)(H(j)) = H(i). That will correspond to a field generated by the fixed fields of all H(i)’s.
Conversely, let F be an intermediate extension of L/K. Then F intersects every finite Galois extension in some finite intermediate extension, yielding a corresponding subprojective limit. These two correspondences are inverses of each other by the fundamental theorem of (finite) Galois theory. Further, the subprojective limit defined by H(i) is a normal subgroup iff each H(i) is normal in G(i), so the second part of the fundamental theorem holds as well.
Obviously, L/F is finite iff the subprojective limit is a finite group, and F/K is finite iff the subprojective limit has finite index in the projective limit. In the latter case, F is contained in some finite Galois extension, so it arises from a single H(i); then for all j with i <– j, H(j) is the preimage of H(i) in G(j), and for all k <– j, H(k) = f(kj)(H(j)). Conversely, a subprojective limit that arises from a single H(i) this way has the property every g in the inverse limit that maps into H(i) in G(i) maps into the preimage of H(i) in each G(j) with i <– j, so that it’s in the subprojective limit; the subprojective limit is then the inverse image of H(i) in the projective limit, so its index in the projective limit is [G(i):H(i)], which is finite.
[...] Original post by Alon Levy [...]
[...] proof before you look up the theorem. Alon Levy raises the level another notch in a discussion of infinite galois theory – a part of his ongoing series on galois theory. Mark ChuCarroll steers us over to the topology [...]
Hi, my sites:e6dcb259de94e1b03d1e2aafd23be664
dehumidifier…
[...]Galois Theory: Infinite Galois Theory « Abstract Nonsense[...]…
Virus Support…
[...]Galois Theory: Infinite Galois Theory « Abstract Nonsense[...]…
My brother suggested I might like this web site.
He was totally right. This post actually made my day.
You can not imagine just how much time I had spent for this
information! Thanks!
Wow, wonderful blog layout! How long have you been blogging
for? you made blogging look easy. The overall look of your website is wonderful, as well as the content!
Asking questions are actually good thing if you
are not understanding something fully, however this post presents pleasant understanding yet.
Nice blog here! Also your web site loads up fast!
What host are you using? Can I get your affiliate link to your
host? I wish my website loaded up as fast as yours lol
You need to be a part of a contest for one of the best
blogs on the web. I most certainly will highly recommend this blog!
Now the time has come to enjoy my present, an electric cigarette from e Health Cigarette,
a surprise from my brother to quit smoking cigarettes.
Cheers!
Excellent blog! Do you have any tips and hints for aspiring writers?
I’m hoping to start my own blog soon but I’m a little lost on everything.
Would you suggest starting with a free platform like WordPress
or go for a paid option? There are so many choices out there that I’m completely overwhelmed .. Any recommendations? Kudos!
Howdy would you mind letting me know which hosting company you’re utilizing? I’ve
loaded your blog in 3 different web browsers and I must say this blog loads a lot faster then most.
Can you recommend a good hosting provider at a honest price?
Cheers, I appreciate it!
I’ve been browsing online more than 2 hours today, yet I never found any interesting article like yours. It’s pretty worth enough for me.
In my opinion, if all webmasters and bloggers made good content as you did, the net
will be much more useful than ever before.
I could not resist commenting. Well written!
Superb page, Maintain the fantastic work. Many thanks.
I was suggested this website by my cousin. I am not sure whether this post is
written by him as no one else know such detailed about my trouble.
You are amazing! Thanks!
I’ve been browsing online more than 3 hours today, yet I never found any interesting article like yours. It’s pretty worth enough for
me. In my view, if all site owners and bloggers made good content as you did,
the net will be much more useful than ever before.

I could not resist commenting. Very well written!

Great website, maybe we can share links?
I always emailed this blog post page to all my associates, for the reason that if like to read it after that my contacts will too.
With havin so much content do you ever run into any issues of plagorism or copyright violation?
My blog has a lot of exclusive content I’ve either authored myself or outsourced but it appears a lot of it is popping it up all over the internet without my permission. Do you know any solutions to help reduce content from being ripped off? I’d truly appreciate it.
Wow, this article is pleasant, my younger sister is
analyzing these things, therefore I am going to let know her.
Hello, Neat post. There is an issue together with your website in web explorer,
could test this? IE nonetheless is the market chief and a
large element of people will omit your magnificent writing because of this problem.
Pretty section of content. I just stumbled upon your site and in accession capital to assert
that I acquire in fact enjoyed account your blog posts.
Anyway I’ll be subscribing to your feeds and even I achievement you access consistently rapidly.
Hi there just wanted to give you a quick heads up and let you know a few of the images aren’t loading correctly. I’m
not sure why but I think its a linking issue. I’ve tried it in two different internet browsers and both show the same results.
Hi, I think your blog might be having browser compatibility issues.
When I look at your website in Firefox, it looks
fine but when opening in Internet Explorer, it has some overlapping.
I just wanted to give you a quick heads up!
Other then that, superb blog!
Hi there, You’ve done an incredible job. I’ll
definitely digg it and personally suggest to my friends.
I’m sure they’ll be benefited from this site.
Are you within the market to get a loan? Do you have credit score?
There are various options accessible. Money advance loans could be a good choice when you are in need of emergency funds
and there is nowhere else to turn. This article will help you with the best
suggestions for obtaining a payday loan.
This is a really good read for me, Should say that that you are among
the best bloggers I actually saw.Thanks for publishing this informative article.
Thanks a lot!
I’m really enjoying the design and layout of your blog. It’s
a very easy on the eyes which makes it much
more enjoyable for me to come here and visit more often.
Did you hire out a designer to create your theme? Outstanding work!
I’m very pleased to find this web site. I need to to thank you for ones time due to this fantastic read!! I definitely liked every little bit of it and i also have you book marked to look at new stuff on your web site.
If some one needs to be updated with hottest technologies afterward he must be pay a quick visit this web site and be up to date everyday.
My wife had arrived from the South, so I was at good hands.
–LINK REMOVED –>Grammy Nominations 2011<. As the atmosphere began to disintegrate around me, I felt a whole new strength forge into my thoughts.
Fantastic beat ! I would like to apprentice while you amend your site, how can i subscribe for a
blog web site? The account aided me a acceptable deal.
I had been a little bit acquainted of this your broadcast offered bright
clear idea
Wonderful article! That is the type of information that are supposed to be shared around
the net. Disgrace on Google for now not positioning this submit upper!
Come on over and discuss with my web site . Thank you
=)
The style of anti snoring pillows varies from pillow to pillow, but all of them have the identical standard characteristics, which have been elaborated by
the sleep disorders’ specialists. If you place your head on these pillows you happen to be encouraged to sleep in your side, or in your stomach, in addition to, it keeps your chin away out of your chest, in order that your jaw is kept extended forward. All this outcomes in enlarged breathing airway, in order that you can find no obstructions in your air passages and you cease snoring.
Hmm it appears like your blog ate my first comment (it was
extremely long) so I guess I’ll just sum it up what I submitted and say, I’m thoroughly enjoying your blog.
I as well am an aspiring blog blogger but I’m still new to the whole thing. Do you have any suggestions for beginner blog writers? I’d genuinely appreciate it.
Eiffel Pillar this guide has aided you! http://www.mycarhiremalaga.
co.uk/ One of the many industries Sc in next 3 sc, 2 sc in adjacent sc
repeat some.
Hi there superb blog! Does running a blog similar to this require a massive amount work?
I have no knowledge of computer programming however I was hoping to start my own blog in the near future.
Anyhow, should you have any recommendations or tips for new blog owners please share.
I know this is off subject but I just had to ask.
Many thanks!
The media for downloading has never been this fast and simple as
it is now. I am not a PC player, please anyone assist me to to remove Dell Inspiron password…” We usually hear such inquiry around
us. Typically, the My Documents folder may be the best
anyone to use.
Quality content is the crucial to be a focus for the viewers to visit the site, that’s what this website is providing.
Volleyball is one of the great outdoor games that thrives on this type of healthy domination.
Next, contain the kids with picture at your fingertips of their newly found vegetable discover the actual vegetable in the
produce portion of your local supermarket. What has catapulted miracle traffic bot
to fame may be the fact that is has been used extensively
in the video game development.
Hello there, I found your blog by the use of Google whilst searching for a related topic, your website got here up, it
looks great. I have bookmarked it in my google bookmarks.
Hi there, simply turned into aware of your blog thru Google, and located that it’s truly informative. I am going to watch out for brussels. I will be grateful should you proceed this in future. Lots of other folks shall be benefited out of your writing. Cheers!
Have you ever thought about including a little bit more than just
your articles? I mean, what you say is important and everything.
But think about if you added some great photos or videos
to give your posts more, “pop”! Your content is excellent but with
images and videos, this website could certainly be one of the
very best in its field. Awesome blog!
I was very pleased to find this website. I wanted to thank you for your time
due to this wonderful read!! I definitely really liked
every part of it and i also have you bookmarked to check
out new things in your blog.
That’s simply because I that Help oneself you relax and then put them into practice to get you through and through this nerveracking clock time. cloudcigs.webeden.co.uk/ In addon, the mental object and timbre of the compress handout suggested that toxic or harmful in the E cigarette cartridges.
Merlin of fees to use their services and and then just offers you a pocketsized component part
of the profit. Web Hosting There are masses who assume offer a
vast compass of instruments from the passing a littler sum of money value.
It cost they are actually pretty affordable for how
popular they are. http://www.synechism.co.uk/ – e cigarettes The dope produced when smoke these items is of two diseases
exhibit in the Affected role.
What’s Taking place i’m new to this, I stumbled upon this I have discovered
It absolutely helpful and it has aided me out loads.
I hope to give a contribution & assist different users like its aided me.
Good job.
Appreciation to my father who told me concerning this blog, this blog is in fact remarkable.
Vedonlynnin kohteena on ottelun unless he is satisfied
with the integral website’s execution and gaming glide path. http://onlinebestcasino13.webeden.co.uk proper now it is possible to get 5 Unloosen taken caution of before one starts qualification deposits and having fun.
It’s going to be ending of mine day, except before end I am reading this great piece of writing to improve my experience.
As the admin of this web site is working, no doubt very quickly it will be famous, due to its
feature contents.
I am more active since I wish to be. Issues: Hydroxycut is pointed
out to set off troubles on the very first few days
of use. As a plant, you could see Garcinia Cambogia as a compact, yellow fruit.
Hello There. I found your blog using msn. This is a really well written article.
I’ll make sure to bookmark it and come back to read more of your useful information.
Thanks for the post. I will definitely return.
After a a long time of searching i’ve finally found a website that gives you
a free combat arms aimbot.
It works 100% perfectly. as far as i’ve used it, I’ve used it
for months, It has been undetectable.
i haven’t gotten banned, and it’s great. No more constant deaths on
combat arms. With this amazing
combat arms aimbot, i get like 10 multi kills every single match, and im able to get my
weapon of choice
a lot faster and easier. If you guys want you can check out the hack
at http://imgur.com/ca09o0D
Wow, this article is nice, my younger sister is analyzing such things,
so I am going to tell her.
Great goods from you, man. I have consider your stuff previous to and
you’re just too fantastic. I actually like what
you have bought right here, certainly like what you are stating and the best way by which you assert it.
You are making it enjoyable and you still care for to stay it sensible.
I can not wait to read far more from you. That is really a
great site.