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I am currently an undergraduate student in math, but this question is already bugging me, because I am about to start an undergraduate research project that will likely become my Masters degree as well.

I learned Algebra on my own, and really seemed to enjoy Galois theory. I've even made an inquiry on Math.SE (Is there active research in Galois theory?) to see if I could still study it more in depth. The thing is, not a single professor on my university works on this area or anything related directly to it. Since I also loved the rest of Algebra, my project will be on something else (ring theory, linked to a research group on the topic), but I still can't shake away the desire to learn more Galois.

So how do I actually go about studying something nobody else seems to be studying at my university? I know this can be done, because every research group began at some point with a first person interested in the topic (in fact, I know about professors who have created their own research groups relatively recently), but I really just don't know what I can do as an undergraduate/early graduate student.

Thanks in advance!

This is sort-of (but not entirely) related to the following thread, which inspired me to write this one:

How realistic it is to change to a different field of mathematics after PhD?

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  • I think my answer at academia.stackexchange.com/questions/145120/… might be relevant – Alexander Woo Jun 13 at 21:27
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    It's not really ideal to be studying a subject at a university where literally no-one else studies that subject. – Tom Jun 13 at 23:49
  • Is your university large enough to offer a masters degree in math, or small enough that you don't have an algebraist that can get you started? In a separate comment, you mentioned some professors helped you set up an algebra sequence. Do they have an opinion on how to study Galois theory? – Teepeemm Jun 14 at 17:35
  • The research topics in the linked answer seems fancy, but to be honest, in my opinion, they are bit above the undergraduate (even masters) level. If you happen to a professor with a knowledge of category theory you can do a decent project on Galois connection. For example Nullstellensatz is a Galois connection in the ring theory context. In fact, this will prepare you better for higher studies. – Bumblebee Jun 14 at 18:10
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    One issue that will arise in talking to mathematicians about whether they do research in "Galois theory", is that the contemporary sense of "Galois theory" is exactly as a reference to the basic theorems that will be found in standard algebra texts. Does not refer to any active research area. Or, maybe "Galois theory of number fields", which then is part of "algebraic number theory", which is an active research area. Or, more fancily, "Galois representations", whether part of Langlands' Programme or not, is a topic of current research. But "Galois theory" does not usually refer to these... – paul garrett Jun 14 at 19:06
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This is harder to do for most undergraduate students since they don't have sufficient experience to get by without feedback on what they study. Normally learning requires both reinforcement and feedback. Reinforcement comes from applying what you study in some way, exercises for example. Feedback normally comes from a professor evaluating what you produce. Both of these seem to be missing, so the task is harder.

But, people can, and do, study and learn on their own, using books, say. Galois theory is well represented in the literature, but you need to find one or more that provide some way for you to apply what you learn. Professors learning a new topic will do this, but they will also seek out others with whom they can raise issues.

Perhaps one of your professors, while specializing elsewhere, can answer your questions as they arise and give you feedback on some of your attempts to solve problems or pose questions. Perhaps they have an external colleague that is more specialized and who they can put you in contact with. That has value in its own right, actually. But you will need to depend, primarily, on existing works, but not on reading alone, which provides too little reinforcement to give deep learning.

Eventually most mathematicians learn to do this. You are starting a bit early and from a smaller base, but it is possible.

Alternatively, focus hard on what you can get the most help with (Ring Theory, say) and delay some of the other topics of interest until you have more experience and the opportunity for more help. You don't need to learn everything this year.

You can also keep a notebook of questions on what you study. Review it periodically to see if you now have answers to those questions. Such a notebook can also be a source of ideas for research as you get more experience.

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  • Thank you so much, I think I get the gist of it. For reinforcement, I have purchased a couple of books specific to Galois theory, in order to delve deeper (maybe not this year, but still good to already own them) - one of which has a lot of exercises to practice. As per feedback, I’ll talk to my professors; many of them have taught undergraduate courses on it, so maybe they can help, at least in the beginning. Again, thanks a lot! – Gauss Jun 13 at 15:32
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    Let me emphasize: it's quite likely that one or more of your professors themselves knows quite enough about Galois theory to advise you on undergraduate/early graduate independent study. There are many more research mathematicians who can do that than there are research Galois theorists. – Lee Mosher Jun 14 at 13:51
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The Short Version:

Question:

How do you study Galois theory at a university where no-one else studies Galois theory?

Answer:

You apply to attend a different university and switch schools after you complete your bachelor's degree


The Long Version

Use Google to find living experts in Galois theory.
Get these people's email address. Talk to them.

The best way to get into graduate school is to find a professor you want to work with. You can ignore most aspects of the school itself.

Also, most professors can sponsor admission of graduate students. I do not mean that professors will pay money, or pay a student's tuition. I mean that after you fill out an application, the office of admissions will say "yes." Your SAT scores, etc... will not matter too much if their is a professor who says, "I want Joe Shmoe to be my student."

Tuition is not usually a problem, because graduate students are paid a stipend to work as either:

  • teaching assistants
  • research assistants.

The higher the tuition, the higher the stipend. For example, Notre Dame is a very expensive private school located in Indiana (There is a different Notre Dame in Ireland) pays graduate students (masters students and Ph.Ds) more than 30,000 US dollars per semester.

After you finish your bachelor's degree, begin studying at your new University.

You cannot transfer credits per se. However, if you finish the baccalaureate degree, then you will not have to transfer credit hours.

If you only need 6 credit hours to complete a bachelor's (baccalaureate) degree, then you will NOT be able complete the remaining 6 credits somewhere other than where you started.

You cannot usually complete a baccalaureate at the University you've barely began studying at.

You do NOT need to transfer credits if you finish the baccalaureate degree. After that, it's like starting from scratch, but for a masters or doctoral degree.

  1. Find an advisor (at a different University) who studies Galois Theory.
  2. Email the Galois Theorist.
  3. Finish your baccalaureate degree at the University where you are now.
  4. leave your Alma Mater and make a new home at the University with people who study Galois theory.

Find a university with more than one expert on Galois theory. That way, you can switch advisors if you hate the first person you work with.

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I didnt know there are universities that do not offer a module on algebra which includes Galois theory, this seems a bit troubling.

Generally: don't do it. Pick only a topic which is known at your university or switch to a different univerisity. Your argument that new research groups are founded is not valid, since this either happens by hiring somebody from somewhere else (where there was a group) or it is a new subject. In a new subject a trained researcher can make progress. In an established subject you cannot. You need guidance about what is an interesting problem on which you can work on and you need to have meaningful discussions.

Also no professor will give a topic for Galois theory if no one is working on that. (Lets assume for a moment somebody will though, then how is anybody grading it?)

Long story short: change topic or university.

PS: just to add: it is great you found a topic which is highly interesting to you.

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    To clarify: there are modules in Galois theory - one course in undergrad level and one in grad level. There is just no researcher who does research in the subject – Gauss Jun 14 at 10:58
  • @Gauss thanks, but then why did you learn it own your own? (This doesnt change my reply though) – lalala Jun 14 at 11:07
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    It’s a long story, but basically, I wanted to introduce myself to the three major areas (Algebra, Geometry and Analysis) and, since I loved algebra so much, I talked to some professors and together we decided I’d study the equivalent of an undergraduate sequence in algebra on my own so I could begin my undergraduate research on a higher level. Said undergraduate sequence included some Galois theory – Gauss Jun 14 at 11:22
  • Hah my undergrad is notoriously bad for algebra. We barely got up to polynomial rings at the end of the one semester of algebra we have. – Cameron Williams Jun 14 at 19:34

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