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This fall I am starting the 3rd year of my bachelor's degree and I intend to continue my studies with a Master's degree and possibly a PhD. The university I study at is a respected institution in my country but not particularly focused on my area of interest (algebra). I had a first contact, at a summer school, with students from top universities in Europe and I realised I am under prepared for a Master's at one of these universities.

I have a good academic record and did some undergraduate research but there are topics I know nothing about. For example in algebra I had only 2 undergraduate courses (vector spaces, groups and rings- basic definition, properties, isomorphism theorems), no linear algebra aside from the matrix of a linear transformation.

So my question is: what courses are essential to someone who wants to go to graduate school in algebra at a good university and what should I study to bridge the gap?

Edit: I already took all the undergraduate and graduate level classes available in my field,at my university, but all those were still under the level of preparation of my peers from other universities. So I was asking for subjects I should know in order to study by myself.

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    Could you say a bit about which countries you are hoping to apply to? (I can give decent advice for the UK, for instance, but very little for e.g. France or Germany) – Yemon Choi Aug 4 '17 at 19:57
  • I have not decided yet. I am interested in modules and category theory but I am not sure what universities are good in this fields. I am considering UK, Germany, France and Canada but I still have to do a lot of research to determine what institution would be a good fit. – Ana Aug 5 '17 at 12:42
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The simple answer is that if you're interested in a subject and expect to continue on to a master's or PhD (or even if you're not expecting to go for an advanced degree, you simply expect to pursue it as a career), you should take as many classes in that subject as you can. Pick the best or most interesting to you from whatever is offered by your department to fill out your electives and do your best.

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Take a look at these oral exams, taken by 1st/2nd year grad students at Princeton:

http://web.math.princeton.edu/generals/

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    Depending on where the OP is intending to apply, I think the Princeton quals are not a very realistic guide – Yemon Choi Aug 4 '17 at 19:50
  • its a top university, isn't it ? – mystupid_acct Aug 4 '17 at 20:12
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    The OP wants to repair deficiencies in their knowledge before applying to do a Master's; this does not mean it is a good idea to look at what much better prepared people will be made to study at a very demanding PhD program with a notorious style of qualifying exam – Yemon Choi Aug 4 '17 at 20:22

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