I am self learner currently learning mathematics. I am studying elementary real analysis from Stephen Abbhot's book and i have some experience with proofs also from book "How to prove it". I would like to embark on journey to abstract algebra. I have looked similar suggestions here and came up with two books which are as :  1. Abstract algebra by Charles Pinter  2. A first course in abstract algebra by J.B.Fraleigh  I would like to know any more books which are available for self study so that i can have a look at them before buying and advice in general to subject.  Thanks
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Does anyone have any suggestions for abstract algebra books particularly suited to self-study?     Here is some background and motivation, if it's helpful.  I'm currently a junior in high school, but I have some familiarity with groups, rings, polynomials and fields, as I went through Fraleigh's book. I liked its writing style, but many of the problems were not helpful. I would like to get better at algebra with some other book, particularly one that's aimed at someone who already has a little familiarity with the objects, but is by no means completely comfortable.  One of my longer-term goals is to learn a bit of algebraic number theory, as I've only been studying elementary number theory. That's why I feel it  good to learn more algebra first. (I'd also be interested in suggestions of other good subjects to learn prior to tackling algebraic number theory.)  I was considering using one of Lang's books, since they seem pretty universal, but I've heard rumors that they are very terse and not good for self-learning.
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I am almost finishing Gilbert Strang's book "An introduction to linear algebra" (plus video lectures at MIT OCW). First and foremost, I would like to suggest this course for everyone. It has been incredibly illuminating.   I would like to continue studying linear algebra, with particular focus on different properties of matrices and the transition to more general linear spaces (I am a physicist so Hilbert spaces and etc. are of particular interest).   Does anyone have a a good recommendation of books/resources/etc.?
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I would like to teach myself measure theory. Unfortunately most of the books that I've come across are very difficult and are quick to get into Lemmas and proofs. Can someone please recommend a layman's guide to measure theory? Something that reads a bit like this , starts out very gently and places much emphasis on the intuition behind the subject and the many lemmas.
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What would be a good book to learn basic number theory? If possible a book which also has a collection of practice problems? Thanks.
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Background: Due to some unfortunate sequencing, I have developed my abstract algebra skills before most of my linear algebra skills. I've worked through Topics in Algebra by Herstein and generally liked his approach to vector spaces and modules. Besides a very elementary course in linear algebra (where most of the time went towards matrix multiplication), I have not developed any other linear algebra skills.  But it seems that it is now necessary for me to do so. Most of the topics that I am looking at now require some background of linear algebra and I still lack understanding of ideas like: bilinear forms, invariant subspaces, eigenvalues, requirements for diagonalization of a matrix and so forth.  This brings me to my question (provided there are any)     Question: What are some good texts that develop the theory of linear algebra from a perspective of general algebra?   Are there any texts that develop the key (elementary) ideas of linear algebra in an abstract setting? Thank you for the help!
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Is there a good book on introduction to Algebraic Topology which is self-contained and does not assume any background in topology, only standard calculus and linear algebra courses?
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I was originally intending on reading Mathematical Logic by Joseph R. Shoenfield (1967) though after reading  I get the impression that despite being influential for the time it was written, there now exist many new books written from our current perspective that tend to be easier on non-experts by offering more thorough explanations for each new topic being introduced.   As of now I have a basic knowledge of abstract algebra (basics on vector spaces, rings, groups etc.) from doing elementary number theory and taking a course on linear algebra, also I have completed  a year long course on real analysis which used a lot of elementary set theory and some order theory.   With all that said, I was hoping based on my current basic knowledge of math that someone could recommend me an introductory book on mathematical logic which I could use as an alternative to Shoenfield's and which addresses Peter Smith's criticisms - any help would be greatly appreciated.
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Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction to Analysis" by Gaughan.   While it's a good book, I'm not sure it's suited for self study by itself.  I know it's a rigorous subject, but I'd like to try and find something that "dumbs down" the material a bit, then between the two books I might be able to make some headway.
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I went through this thread on  asking about a text on number theory, but my question is slightly different.   I was looking for a short and sweet book on elementary number theory. I don't need it to be a definitive treatise on number theory. I just need to have book which covers enough material to advance to algebraic number theory. I have looked at the books by Niven-Zuckerman-Montgomery and the Jones' and the one by Burton. All of them were very readable and I would like to read at least one of them at some point. For now, I would just like a book (say 100 pages long) which can get me acquainted with most of the material needed for algebraic number theory. Basically there is a course on algebraic number theory that I wish to take which begins in a month's time and I have not done enough elementary number theory.
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I'm a looking for a good book to self-study differential forms. Particularly, I'm looking for a book that is as similar as possible to Bert Mendelson's "Introduction to topology" (i.e. a book that procede by  following a: "Definition, theorem, proof" style). In addition, the book that I'm looking for should be as much self consistent as possibile. I'm a first year graduate student in nuclear engineering. My prerequisites are a good understanding of (multivariate and vector) calculus,  linear algebra, and a little of functional analysis, Lebesgue integration theory, PDE. I know nothing about differential geometry, but to my (very) poor understanding  differential forms and concept like manifolds and so on are linked to each other.
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I searched on this site but I found posts asking for books on complex analysis which I guess is not what I want.   I want a book on problems concerning with roots of unity, argand plane, exponential form, inequalities etc. Some examples : , , .  I don't think these types of problems are covered in a analysis book but I don't know. I think I want a book on algebra of complex numbers.   Are these topics covered in analysis book ? If not, please suggests some books on this topic. I guess my question is bit vague, sorry for that. :).      As suggested in the comments I would like to have books similar to .
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I'm currently working my way through Harvard's online abstract algebra lectures (if you're interested, you can find them ).  The lectures come complete with notes and homework problems.  Of course, since I don't actually go to Harvard, I can't hand in the homework assignments to find out if I'm doing them correctly.  So I've decided to try to crowd source the grading of my solutions.  I'll post individual questions as I finish them and wait for comments.  I would like to get critiques of not just my reasoning but the style of my write ups as well.  Also, any alternative approaches to the problem would be welcome.  I've looked for forums dedicated to these kinds of OCW courses but have not been able to find any.  This surprises me.  It seems like, with the advent of these free online educational resources, an online meeting place for those who take advantage of them would be a natural offshoot.  So this might be a bit of an experiment.  Unless I'm the 873rd person to post something like this here.  If that's the case, sorry for being so long winded.  Anyway, on with the question.  This one is assigned in the 4th lecture.     Let $V$ denote the Klein 4-group.  Show that $\operatorname{Aut}(V)$ is isomorphic to $S_3$.   (I've moved my solution below to keep this question from showing up in the unanswered list.)  Thanks...
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I'm starting to read Baby Rudin (Principles of mathematical analysis) now and I wonder whether you know of any companions to it. Another supplementary book would do too. I tried Silvia's notes, but I found them a bit too "logical" so to say. Are they good? What else do you recommend?
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I want to self-study some abstract algebra and cryptography during the summer, so what are some of books that are suitable for self-study? I have very limited background in algebra and none in cryptography, and like to learn by examples. Also, please let me know if there are any books on these subjects with solutions to most of the exercises, so I can check on myself. (I saw 's post about good algebra books, but I would like to focus more on those with good exercises and detailed solutions.)  All suggestions are welcome! Thank you!!
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sorry for asking the same question in a slightly different angle, I want a book in Category Theory similar to Dummit and Foote's book in Abstract Algebra. I want it to have tons of examples and problems, and I want it to be the sort of book in which you always have an idea about what the next results are going to be. Something I really liked about Dummit and Foote is I understood what was going on the first time I read it, but when I read it a second time I felt a lot better than the first time (I'm looking forward to a third).  Sorry if this just sounds like a lost of demands, I just wanted to give some information about what I was looking for.  Thank you kindly  Regards.
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I am currently undergoing a full Apostol Calculus I reading, and with few chapters left I've began planning what other books to read. I was thinking about finishing both Apostol Calculus l and ll, and then read Apostol mathematical analysis. I'm not pretty sure whether these are the best books or if they will provide me with a strong math basis. I've already taken calculus i, ii, iii and linear algebra. The reason I've chosen Aapostol volume 1 and 2 is that I've heard they cover almost all the topics I've seen and will see in the next semester/s. I would appreciate if somebody could give me some advice about this. I really like to have a book that really gives me hard and challenging problems. I don't know whether to read Spivak Calculus book or not.
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I need a graduate level, mathematically rigorous textbook on multivariate analysis and inference to enhance myself. I've been reading the  and doing the problems within, but I need a book with other focuses. Topics like famous distributions (Wishart, Wilks lambda, etc.), hypothesis testing, some theory on estimation (point, interval), and other more modern materials are welcome. I've checked , but the OP there was looking for something useful for his psychological analysis, which is not exactly what I want here. Currently I have the , so I would also like to hear the comments about this book and its exercises. Thank you.