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I am a second year undergraduate and I'll be appearing this exam in about a year and half. This is the link to the question paper of PhD and Integrated PhD program: Question Paper 2017 . It contains Algebra, Analysis and some Topology.

I was looking for textbooks which will help me prepare and at least attempt 50% of the problems in the question paper. Books I have in mind are Dummit and Foote for Algebra and Rudin for Analysis but I feel some of the questions asked in the paper are beyond those textbook. I'll appreciate it if you suggest me some books and guide me as I intend to work alone.

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It seems to me that the actual material that you need to know on these questions would probably be covered by those textbooks.

The problem is that to do well on an exam like this, you need to know that material very well. Most people acquire that knowledge through repeated exposure to the material beyond the textbook they learned it in, either by working on problems that involve it, or learning more advanced material based on it.

Once you're satisfied that you know the material from those textbooks, you could spend time on problems. In analysis, you might use a problem book like the one by Makarov or the three-volume book by Kaczor and Nowak.

In algebra, I'm less aware of what problem books there are that would be at the right level. It might be preferable to instead learn the material from a more advanced textbook than Dummit and Foote, such as the two-volume one by Jacobson, provided you are able to handle the additional difficulty.

I would also say that the problems on the sample exam appear to be of a similar level of difficulty to the easier problems in the Putnam competition, or to many of the problems on the Berkeley prelim, so problem books intended as preparation for those exams might be useful.

https://www.springer.com/gp/book/9780387204291

https://www.math.hmc.edu/putnam/

https://www.amazon.com/Selected-Problems-Translations-Mathematical-Monographs/dp/0821809539/

https://www.amazon.com/Problems-Mathematical-Analysis-Student-Library/dp/0821820508/

https://www.amazon.com/Basic-Algebra-Second-Dover-Mathematics/dp/0486471896/

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