I would like to do a graduate degree in mathematics, and I have a full year before I will be able to do so (for personal reasons). I mainly have my weekends available to study.
I am interested in Abstract Algebra in general, I really liked courses like groups, rings, field theory/Galois theory etc'.
I have always found analysis to be difficult for me and courses in the scope of Abstract Algebra felt more natural and intuitive (though I did have some difficulty with some of the more advanced parts of those courses).
I was told that other then taking some credit points I will get a question from the supervisor, one that is not solved yet but seems at a reasonable difficulty to him, and I will have and I will have to try and find a solution for it.
I would like to prepare myself, in this year, to have control over varies subjects in Abstract Algebra so that I will be more likely to be able to solve such a question (or even understand it, as there are some subjects in Abstract Algebra that I have not learned in any of the courses I took as an undergraduate student).
Also, I have lately been aware of some subjects that involve both analysis and Abstract Algebra, such as topological groups.
I would be happy if I could avoid such topics, but I don't know what type of mathematics is studied in a graduate degree level, so this leads me to the following questions:
What topics of Abstract Algebra should I study in depth ? what topics in Abstract Algebra should I be familiar with their basics ?
Are there any topics in analysis, topology etc' that are likely to be needed for answering a graduate degree level type of questions ?
What should be the focus of my work, should I try to do many exercises within the text, or focus on the proofs and the theory ?
I have the book Abstract Algebra by Dummit and Foote to study with, as well as books in other area of mathematics such as Topology by Monkers that might help me with this goal.
I would be extremely grateful for hearing opinions and advices on this matter!
Note 1: I would like to mention that although I try to avoid analysis, I still had to take courses in that, so I don't lack elementary knowledge in many topics, I have taken: Introduction to functional analysis, Real analysis (measure theory), complex analysis, ODE, Introduction to numerical analysis, Introduction to probability theory (the course didn't talk about $\sigma$ algebras and etc' but we did talk about random variables, CLT, etc').
But I don't consider myself to be good at those topics (except maybe probability that I really liked), I understand them, but I am about average at them, so I don't expect that I would be able to do something non-trivial at those topics.
I would like to extend my questions to include the complement of my question to what to study for what I shouldn't spend my time on:
4) Are there topics in Abstract Algebra, or other in other areas that I would need to know (maybe topology ?) that I can skip some parts of (mainly non-core topics that are hard to learn) since they would probably not help me (and due to lack of time) ?