I am currently a junior in a completely unknown liberal arts college/mathematics department. So far, I have take two courses in Abstract Algebra and group theory, one in linear algebra, currently taking Real Analysis. Next semester, I will most probably be taking Complex Analysis,Rings and Fields theory, Introductory Topology and Introductory Number Theory at a nearby larger school. Besides these, of course, I have calculus sequence, probability and statistics, graph theory and Combinatorics under the belt.
So far, I find Algebra extremely fascinating (I will be taking my third course in Abstract Algebra next semester) and its application to number theory (I have self-learned a little about Galois Extensions but not significantly much). Ideally, I would like to go to a nice enough program where I can learn and do Algebra and Number Theory. However, I do not have any idea as to how hard it is to get into a school with such programs. I searched over the internet and could only find like topmost school that offer Number Theory. I do not think my profile is competitive enough to get into any of such programs. I have not done any summer research yet, except one that I did at my home institution (which bordered more on engineering than mathematics). I have gotten a scholarship at my home institution, which is like the largest one here, but it is not something huge as seen by an outsider. My teachers are not working mathematicians at all. They completed their PhD some 20 years ago, and stopped doing any mathematics except teaching. So I don't know how much their recommendations will count. I do not have anything Earth-shattering to show for in my profile either.
How likely would it be for me to get into a program where I can have an option of doing Number Theory and/or Algebra? I would also appreciate if you could tell me about such programs (except ones that are already listed in US NEWS AND WORLD REPORT). Any other general advice would be highly appreciated too. Thank you in advance.
If you've already taken a couple courses in Abstract Algebra, try reading dummit and foote and do the hard problems. As far as number theory books go, I'm not sure since I have not taken any Number Theory at my University.
I used to look at the syllabi and homeworks for classes at top schools and also found them very difficult. Even at those schools, only the top math majors make it through Analysis and Topology so don't be discouraged. If possible try getting an independent study.