I would like to know some good problem books in various branches of undergraduate and graduate mathematics like group theory, galois theory, commutative algebra, real analysis, complex analysis, topology etc. The books should contain solution to many of the problems if not all, something similar to Problems in Algebraic Number Theory by M. Ram Murty, Jody Esmonde. Also, any text book contain solutions to many of exercises will also do.

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@Mohan They felt like it. Trust me,they don't need a reason........ –  Mathemagician1234 Feb 18 '13 at 6:15
Most math textbooks do not contain solutions to exercises because doing so encourages bad study habits. If you struggle over a problem and frustrate with it, then when you finally get it, it's yours. That result will stay with you, as well as the technique that you developed to get the solution - not anybody else's. If you get completely stuck, ask a professor (or us) for a hint. It's bad to have the option to just flip back and read a solution whenever you don't understand something. In other words, I don't like the premise of this question. –  Alexander Gruber Feb 18 '13 at 6:33
@Alexander I have to most strongly disagree with you on this. I think every textbook should have complete solutions and the reason is it's not enough to know the answer's wrong,you have to know WHY it's wrong and a beginner may have trouble grasping this. It's the STUDENT'S responsibility not to abuse the use of the solutions-they should be consulted only when all else fails.Even better would be a manual made up entirely of hints-a good, well worded hint is usually more then enough for a hard working student to get unstuck. –  Mathemagician1234 Feb 18 '13 at 6:42
@Alexander: Some very good textbooks contain solutions to at least a large number of questions: Graham, Knuth, & Patashnik, Concrete Mathematics, Miklós Bóna, Introduction to Enumerative Combinatorics, and Herbert S. Wilf, generatingfunctionology come to hand at once. In my experience such books are to be preferred for self-study. –  Brian M. Scott Feb 18 '13 at 7:32
For some bizarre reason this question currently has $3$ votes to close, at least two of which are apparently on the demonstrably specious grounds that it is ‘[i]t's difficult to tell what is being asked here’, and ‘[t]his question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form’. –  Brian M. Scott Feb 18 '13 at 7:38

The list of books that I would recommend to you,

Problems and solutions in mathematics, This book contains a number of questions and solutions on topics such as Group and Galois Theory.

General Topology:

1. V. Arhangel'skii, Fundamentals of General Topology .

2. K. Rao, Topology.

Differentiable Manifolds:

1. P. Gadea and J. Masqué, Analysis and Algebra on Differentiable Manifolds.

2. S. Morita, Geometry of Differential Forms.

Mathematical, Real and Complex Analysis:

1. D. Aliprantis and O. Burkinshaw, Problems in Real Analysis.

2. W. Kaczor, Problems in mathematical analysis.

3. R. Shakarchi, Problems and solutions for Complex analysis.

4. B.Makarov, Selected problems in real analysis.

and Schaum's Outline of Theory and Problems are a series of supplementary texts as problems and solutions.

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I second Schaum's Outline. They are very helpful...to this day. –  Fixed Point Feb 18 '13 at 7:41
A very good list,+1. But I'd add that you have to be careful with Schaum's-some are excellent and some are just awful. For example,thier outline on abstract algebra is practically unreadable. So is the one on partial differential equations. –  Mathemagician1234 Feb 18 '13 at 20:20
@Mathemagician1234 , Yes, You're right. Academic level of some of the books in this series are lower than in other books. But i believe that Some of these books are very useful to start (Elementary level). For example, General Topology. –  M.Sina Feb 19 '13 at 4:39

Springer has an entire series titled Problem Books in Mathematics. The two by Lam are well known and have solutions. I don't know if all the books do.

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I mentioned these in a comment, but they might as well be added to the collection in the answers. Graham, Knuth, & Patashnik, Concrete Mathematics, Miklós Bóna, Introduction to Enumerative Combinatorics, and Herbert S. Wilf, generatingfunctionology, all include solutions to a great many of the problems.

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A Collection of Problems on Complex Analysis by Volkovyskii et al. (Dover) takes problems from several well-known texts and furnishes answers for many. I am finding it helpful--problems range from very basic to pretty advanced. It may not be part of the Problem Book series Jim mentions.

And it's inexpensive.

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You may consider Problem Solving Trough Problems by Loren C. Larson. That book is aimed at the advanced undergraduate level, and cover some of the branches of mathematics you need.

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You can begin with the Red and Green books of mathematical problems,both available through Dover. Then there's a ton of good ones,mainly on analysis and combinatorics. A classic is the 2 volume Problems in Analysis by Polya and Szego. Very difficult,but well worth the effort. Before that, you might want to cut your teeth on the wonderful The Cauchy-Schwarz Master Class by J. Micheal Steele, to learn many of the basic inequalities that will be needed there.

After that,the sky's the limit. From T.S. Blyth's 5 volume problem course on abstract algebra to Combinatorial Problems and Exercises by Laszlo Lovasz to the 2 volume Problems in Mathematical Analysis by W. J. Kaczor and M. T. Nowak.

Get pen and paper and get started!

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There is a great list maintained online on http://www.mathpropress.com/mathBooks/. It contains huge list of problem books in most branches of mathematics, even those especially for physicists.

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