(Part 2) Reddit mentions: The best abstract algebra books

I started from scratch on the formal CS side, with an emphasis on program analysis, and taught myself the following starting from 2007. If you're in the United States, I recommend BookFinder to save money buying these things used.

On the CS side:

• Basic automata/formal languages/Turing machines; Sipser is recommended here.
  
• Basic programming language theory; I used University of Washington CSE P505 online video lectures and materials and can recommend it.
  
• Formal semantics; Semantics with Applications is good.
  
• Compilers. You'll need several resources for this; my personal favorites for an introductory text are Appel's ML book or Programming Language Pragmatics, and Muchnick is mandatory for an advanced understanding. All of the graph theory that you need for this type of work should be covered in books such as these.
  
• Algorithms. I used several books; for a beginner's treatment I recommend Dasgupta, Papadimitriou, and Vazirani; for an intermediate treatment I recommend MIT's 6.046J on Open CourseWare; for an advanced treatment, I liked Algorithmics for Hard Problems.
  
  On the math side, I was advantaged in that I did my undergraduate degree in the subject. Here's what I can recommend, given five years' worth of hindsight studying program analysis:
  
  
• You run into abstract algebra a lot in program analysis as well as in cryptography, so it's best to begin with a solid foundation along those lines. There's a lot of debate as to what the best text is. If you're never touched the subject before, Gallian is very approachable, if not as deep and rigorous as something like Dummit and Foote.
  
• Order theory is everywhere in program analysis. Introduction to Lattices and Order is the standard (read at least the first two chapters; the more you read, the better), but I recently picked up Lattices and Ordered Algebraic Structures and am enjoying it.
  
• Complexity theory. Arora and Barak is recommended.
  
• Formal logic is also everywhere. For this, I recommend the first few chapters in The Calculus of Computation (this is an excellent book; read the whole thing).
  
• Computability, undecidability, etc. Not entirely separate from previous entries, but read something that treats e.g. Goedel's theorems, for instance The Undecidable.
  
• Decision procedures. Read Decision Procedures.
  
• Program analysis, the "accessible" variety. Read the BitBlaze publications starting from the beginning, followed by the BAP publications. Start with these two: TaintCheck and All You Ever Wanted to Know About Dynamic Taint Analysis and Forward Symbolic Execution. (BitBlaze and BAP are available in source code form, too -- in OCaml though, so you'll want to learn that as well.) David Brumley's Ph.D. thesis is an excellent read, as is David Molnar's and Sean Heelan's. This paper is a nice introduction to software model checking. After that, look through the archives of the RE reddit for papers on the "more applied" side of things.
  
• Program analysis, the "serious" variety. Principles of Program Analysis is an excellent book, but you'll find it very difficult even if you understand all of the above. Similarly, Cousot's MIT lecture course is great but largely unapproachable to the beginner. I highly recommend Value-Range Analysis of C Programs, which is a rare and thorough glimpse into the development of an extremely sophisticated static analyzer. Although this book is heavily mathematical, it's substantially less insane than Principles of Program Analysis. I also found Gogul Balakrishnan's Ph.D. thesis, Johannes Kinder's Ph.D. thesis, Mila Dalla Preda's Ph.D. thesis, Antoine Mine's Ph.D. thesis, and Davidson Rodrigo Boccardo's Ph.D. thesis useful.
  
• If you've gotten to this point, you'll probably begin to develop a very selective taste for program analysis literature: in particular, if it does not have a lot of mathematics (actual math, not just simple concepts formalized), you might decide that it is unlikely to contain a lasting and valuable contribution. At this point, read papers from CAV, SAS, and VMCAI. Some of my favorite researchers are the Z3 team, Mila Dalla Preda, Joerg Brauer, Andy King, Axel Simon, Roberto Giacobazzi, and Patrick Cousot. Although I've tried to lay out a reasonable course of study hereinbefore regarding the mathematics you need to understand this kind of material, around this point in the course you'll find that the creature we're dealing with here is an octopus whose tentacles spread in every direction. In particular, you can expect to encounter topology, category theory, tropical geometry, numerical mathematics, and many other disciplines. Program analysis is multi-disciplinary and has a hard time keeping itself shoehorned in one or two corners of mathematics.
  
• After several years of wading through program analysis, you start to understand that there must be some connection between theorem-prover based methods and abstract interpretation, since after all, they both can be applied statically and can potentially produce similar information. But what is the connection? Recent publications by Vijay D'Silva et al (1, 2, 3, 4, 5) and a few others (1 2 3 4) have begun to plough this territory.
  
• I'm not an expert at cryptography, so my advice is basically worthless on the subject. However, I've been enjoying the Stanford online cryptography class, and I liked Understanding Cryptography too. Handbook of Applied Cryptography is often recommended by people who are smarter than I am, and I recently picked up Introduction to Modern Cryptography but haven't yet read it.
  
  Final bit of advice: you'll notice that I heavily stuck to textbooks and Ph.D. theses in the above list. I find that jumping straight into the research literature without a foundational grounding is perhaps the most ill-advised mistake one can make intellectually. To whatever extent that what you're interested in is systematized -- that is, covered in a textbook or thesis already, you should read it before digging into the research literature. Otherwise, you'll be the proverbial blind man with the elephant, groping around in the dark, getting bits and pieces of the picture without understanding how it all forms a cohesive whole. I made that mistake and it cost me a lot of time; don't do the same.

If you like the online course lectures, you should definately look at those. I know tons of great schools such as Yale, UCLA, MIT, Stanford etc. etc. offer full lecture series on youtube. Usually the syllabi are online for you to look at so you can get a feel for it.

I am more of a book learner myself so I will try to make some recommends, but when looking for books try googling, reading stackexchange posts and Amazon reviews.

I'm going to disagree with /u/Orion952 on Fraleigh's book, its an alright book but I have seen much better. For Abstract Algebra, I would recommend Nicholson's book. Its a very gentle introduction to the subject. There are lots of computation problems as well as proofs you can work through so you can get a nice feel for the subject. I would also hunt down the pdf for Dummit and Foote's book as well, I thought it was pretty gentle for the most part as well as comprehensive.

For analysis and topology, I have encountered some decent books.

Strichartz for analysis is very wordy and conversational, so I didn't care for it myself hence didn't read very much of it (I much prefer the style of Walter Rudin) but it might be good for starting out.

Bhatt has written a very nice book for analysis and covers a lot of material on metric space topology. I actually know the author pretty well so if you are interested in the book I may be able to hook you up.

Simmons has written a book that has a pretty conversational style, but I wasn't a big fan of his style. Bhatt's book will have a more "traditional" approach, but thats not to say it isn't readable. The first half of the book will cover the same stuff Bhatt's book does and the second half will be more advanced stuff including some concepts from Functional Analysis (which is a pretty interesting topic).

For Topology, if you have read some of the analysis books above, I would say Munkres' book is nice and it has tons of examples. But try googling beginner topology books if you want to get into the subject sooner, I know I have seen a few stackexchange threads on this.

These are really the topics one needs to know to really dive into mathematics beyond rote computation. I'm sure there are more books out there but these come off my head at this moment.

You can try Hungerford's Introduction to Abstract Algebra or Schaum's Outline of Modern Abstract Algebra. The former is very clearly written and great for self-study, but provides a thorough introductory course that may be much more intensive than you are interested in. The latter covers (in a less rigorous fashion) the principles of linear algebra, number systems and modern algebra. It is exceptionally easy to read and understand, but is not much of a textbook, and lacks most of the depth of the first text. To be honest, it is more of a guide to the foundations of algebra.


If you are intent on studying abstract algebra intensively, I would recommend the first book. However, it appears to me that the second text would definitely be more up your alley, as a stronger foundation in algebra will be invaluable when your student is ready to pursue this subject in a more comprehensive way.

Yeah I linked the Algorthmics Lab more for what they are doing, not so much the general courses. I lost track of what became of Wolfram's ideas, so it's great to find the resource. So entrenched in Kolmogorov complexity theory though ....... ugh. I find Kolmogorov particularly challenging.

I think both top-down and bottom-up are essential. Understanding the landscape of fields goes hand-in-hand with hammering through fundamentals. Even just quick bibliography -> abstract -> conclusion skimming through overview paper or papers from citation lists flowing off seminal papers will do wonders to help put fundamentals in context. Google Scholar is your friend. lol. But I guess getting 200-300 level math nice and solid really is the first step.

The Witness!! Great game. It's interesting, I was thinking about building mental models the other day, how Feynman talked about it, and how I should be cultivating that more. I think that's why interdisciplinary work is so interesting, so much can be discovered there (the alchemy plays out differently, la), and it's due to the mixing of different mental models.

Choosing one book is tough. The foundations are analysis and algebra, and it was my introduction to those topics that grabbed me. This brilliant self-contained analysis book starts with the algebraic axioms of the real numbers, and ends with Lebesgue integration. (The other analysis book at this level would be baby Rudin). This book goes from nothing to everything you need to know about series of functions, inner product spaces, and fourier series to tackle pretty much any higher level textbook.

https://www.amazon.ca/Foundations-Mathematical-Analysis-Richard-Johnsonbaugh/dp/0486477665/ref=sr_1_1?keywords=pfaffenberger&qid=1558986484&s=books&sr=1-1

As for algebra, I had the luck of being taught by the author of this concise textbook (Dummit & Foote would be the other one to look at):

https://www.amazon.ca/Abstract-Algebra-Introduction-Groups-Applications/dp/9814730548/ref=tmm_pap_swatch_0?_encoding=UTF8&qid=1558986393&sr=8-1

There seems to be generally agreed upon seminal texts for every field of study. Knuth for compsci, Rudin for analysis, Hatcher for algebraic topology, Dummit & Foote for abstract algebra, Feller for probability, Bollobas for modern graph theory, etc etc.

Out of curiosity, may I ask the books you have lined out? Once you start to choose a specific topic of study, the options tend to become exponential ...

I don’t know of any one book on field theory that I would recommend without qualification. All of them seem to drop one or two aspects that are, to me at least, very important.

My pet peeves are books that tip-toe around the concept of separability without actually showing how it is used. In particular, a lot of books do not show that if a is separable and algebraic over k, then k(a) is a separable algebraic extension.

With that I mind, you should look for a book that does:

Galois Theory and includes separability as an integral part of the treatment.

Algebraic closures

The Primitive Element theorem

Newton’s Theorem (Every symmetric polynomial in n variables is a linear combination of elementary symmetric polynomials in n variables.)

Compass and ruler constructions - for historical context

Solvability of third and fourth degree equations – for historical context

The equivalence of solvability by radicals and solvability of Galois groups.

An introduction to finite fields and an introduction to cyclotomic extensions

Some stuff about traces and norms

Even if you do find a book with all of these things but you still might find it insufficient because you do not understand the author’s way of treating the above topics. It is always possible to run into authors who will treat some subject in such a way that you will become confused about a theorem that you already knew. So I have looked at lots and lots of books with different treatments. Here is a list of some of the books I own and why I own them.

Ian Stewart, Galois Theory – Good introduction but tip-toes around separability.

Joseph Rotman, Galois Theory - Good introduction but tip-toes around separability.

Paul McCarthy, Algebraic Extensions of Fields – Best treatment of separability I have seen. Does not do historical material.

Patrick Morandi, Field and Galois Theory – Seems to do everything

Jean Pierre Escoffier, Galois Theory - Good historical content but treats separability as an afterthought.

I Martin Isaacs, Algebra A Graduate Course – Treatment of fields within a larger context. Does not solve cubic or quartic

Irving Kaplansky, Fields and Rings – Includes lectures on rings and homological dimension.

There are a couple of others that are worth having around for specific purposes.

Harold Edwards, Galois Theory – I think this book is really old fashioned but sometimes old fashioned is very much worthwhile. In particular, the book contains an English translation of Galois’s original memoir.

Gary L Mullen and Carl Mummert, Finite Fields and their Applications – Shows applications of finite fields in combinatorics and cryptography

I do not care for Artin’s treatment of Galois Theory. He starts out with a field and a group of automorphisms of that field. He then considers the fixed field of the automorphism group as the base field. I have two problems with this. First, this setup brings along the normality of the extension for free and makes proofs a lot easier. Second, this is almost the never way problems start. Problems usually start with a base field and a polynomial over it. Then you must construct a normal extension of the field to get anywhere. Kaplansky does much of the same thing but he seems, to me at least, to be easier to understand even though he is very terse.

If I had to do anything, I would start with Rotman and shift over to McCarthy when I wanted to learn about separability. I would keep Isaac’s around for a general reference. I would also look into Edwards and Escoffier for historical context. However, you are not me so I would expect you to have differing opinions.

As you can probably tell, I had a particularly difficult with Galois theory, and I have spent a long time trying to understand it. My advice is to keep your mind open and try to understand it from several different viewpoints and from several different sources.

Edit: sppeling

Hm. It sort of depends on the amount of algebra you already know.

I've heard good things about Gordon James' "Representation and Characters of Groups", but I have no personal experience with it. I've gone through his Representation theory of the symmetric group monograph, and I liked it a lot.

I really like Etingof's Introduction to Representation Theory, as it covers representations of groups and algebras ( also, there is a nice proof of the double centralizer theorem / Schur-Weyl duality ).

For representations of Lie algebras, Fulton and Harris is a good book. Also good are: Humphreys' Semisimple lie algebras and representations and Kirillov Jr's Introduction to Lie groups and lie algebras.

If you have sufficient knowledge of algebra, a good book is
http://www.amazon.com/Representation-Associative-Algebras-Chelsea-Publishing/dp/0821840665/ref=sr_1_1?ie=UTF8&qid=1369686499&sr=8-1&keywords=representations+of+groups+and+algebras+reiner

Finally, maybe I'll say that I started learning representation theory through learning the representation theory of the symmetric group. A good book for this is (mainly chapters 1,2,4). The first chapter gets you through some of the very basics of finite group representations.

http://www.amazon.com/Symmetric-Group-Representations-Combinatorial-Mathematics/dp/0387950672/ref=sr_1_1?s=books&ie=UTF8&qid=1369686564&sr=1-1&keywords=sagan+representation+theory

Usual hierarchy of what comes after what is simply artificial. They like to teach Linear Algebra before Abstract Algebra, but it doesn't mean that it is all there's to Linear Algebra especially because Linear Algebra is a part of Abstract Algebra.

Example,

Linear Algebra for freshmen: some books that talk about manipulating matrices at length.

Linear Algebra for 2nd/3rd year undergrads: Linear Algebra Done Right by Axler

Linear Algebra for grad students(aka overkill): Advanced Linear Algebra by Roman

Basically, math is all interconnected and it doesn't matter where exactly you enter it.

Coming in cold might be a bit of a shocker, so studying up on foundational stuff before plunging into modern math is probably great.

Books you might like:

Discrete Mathematics with Applications by Susanna Epp

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Building Proofs: A Practical Guide by Oliveira/Stewart

Book Of Proof by Hammack

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al

How to Prove It: A Structured Approach by Velleman

The Nuts and Bolts of Proofs by Antonella Cupillary

How To Think About Analysis by Alcock

Principles and Techniques in Combinatorics by Khee-Meng Koh , Chuan Chong Chen

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) by Carol Ash

Problems and Proofs in Numbers and Algebra by Millman et al

Theorems, Corollaries, Lemmas, and Methods of Proof by Rossi

Mathematical Concepts by Jost - can't wait to start reading this

Proof Patterns by Joshi

...and about a billion other books like that I can't remember right now.

Good Luck.

The Galois theory class I took used this textbook: A Course in Galois Theory by DJH Garling. It was probably the first 'high-level' textbook I used and I found it difficult and dense, but very rewarding. It's just something you'll have to get used to. I was able to read it and do decently in the course after only completing a rings and field course and I was taking group theory concurrently. If you have rings/fields/groups knowledge, get it. Although I'm not sure what the overall opinion is on this textbook in the mathematical community. But I'd wager if someone as brilliant as my professor used it, it must be pretty good.

If you're doing both applied and pure abstract algebra rather than elementary algebra, then you'll probably need to learn to write proofs for the pure side. I recommend Numbers, Groups, and Codes by J. F. Humphreys for an introduction to the basics and to some applied abstract algebra. If you need more work on proofs, the free Book of Proofs can help, and Fraleigh's A First Course in Abstract Algebra is a good book for pure abstract algebra. If you want something more advanced, I recommend the massive Abstract Algebra by Dummit and Foote.

There's no single book that's right for everyone: a suitable book will depend upon (1) your current background, (2) the material you want to study, (3) the level at which you want to study it (e.g., undergraduate- versus graduate-level), and (4) the "flavor" of book you prefer, so to speak. (E.g., do you want lots of worked-out examples? Plenty of exercises? Something which will be useful as a reference book later on?)

That said, here's a preliminary list of titles, many of which inevitably get recommended for requests like yours:

1. Undergraduate Algebra by Serge Lang
   
   
2. Topics in Algebra, 2nd edition, by I. N. Herstein
   
   
3. Algebra, 2nd edition, by Michael Artin
   
   
4. Algebra: Chapter 0 by Paolo Aluffi
   
   
5. Abstract Algebra, 3rd edition, by David S. Dummit and Richard M. Foote
   
   
6. Basic Algebra I and its sequel Basic Algebra II, both by Nathan Jacobson
   
   
7. Algebra by Thomas Hungerford
   
   
8. Algebra by Serge Lang
   
   Good luck finding something useful!

Rotman's Advanced Modern Algebra is great. Like Lang it goes far beyond the material normally covered in even a graduate course so you wont likely outgrow it any time soon (if ever). And unlike Lang the early material on group theory is actually presented in a way that is comprehensible to someone who doesn't already know the subject.

On the down side his treatment of fields and Galois Theory is broken up into two disparate sections of the book which I don't like. For that material you just can't find a better source than Lang. But no book is perfect and for a book that can be used as an introduction but also as an advanced text I think the Advanced Rotman is definitely what I would recomend.

It depends on your level, but for a good book that starts with category theory and covers some topos theory, I like McLarty's Elementary Categories, Elementary Toposes. Since Sheaves in Geometry and Logic was covered, I've heard Categories, Allegories is quite good (though I haven't read any of it). Admittedly, I'm new to and am still learning category theory and type theory, so I'm not exactly the best resource on what to read.

Knapp's book is good, but it is quite advanced and not self contained. It assumes a course in Lie groups, ("as in Chapter IV of Chevalley [1946]"). One of my favorite books, though.

Here's a brief outline plus some resources:

First thing to know is that since simple groups have no normal subgroups, they cannot be the kernel of any non-trivial have a non-trivial kernel for any morphism (since the kernel of a morphism is normal). This means that any morphism from a simple group G, is either 1-1 or it's the trivial map. As a consequence, simple groups are in some sense "atomic", you can't map them to a smaller group unless you completely crush them. As such they serve as the "building blocks" of all other groups. If you could classify all such groups, you would have a list of all the "atoms" of finite group theory.

Around 1910, Burnside conjectured that all non-abelian finite simple groups must have even order (I don't have a reference for this). This was eventually proved by Feit-Thompson) in 1962. Around 1970, Danny Gorenstein announced a program for the classification of all non-abelian finite simple groups. In 1983, he announced that the proof, carried out by hundreds of mathematicians and spanning many thousands of pages, was complete. Then, being the only one with the full picture of the proof, he promptly died.

Here in an intro written for physicists, but very useful.

For a more serious exposition (i.e. grad school level class) you might see Wilson's The Finite Simple Groups.

To get a feel for the classification proof and the way you might apply the classification see Stephen Smith's talk

For just the constructions of the groups you might look here. It's worth noting that when giving a "construction", one is looking for a sort of "explanation" of the group (other than giving a list of permutations), i.e. Group G occurs as the automorphism group of object X.


EDIT: For the Monster, the "natural" object on which it acts is a vertex algebra and you can see the construction of the one in question in "Vertex operator algebras and the Monster" by I. Frenkel, Lepowsky, and Meurman

Seriously, that book is awesome. It seems like, for a while, my CS professors were able to pick whatever books they wanted in teaching their classes. So we had theory of computation with Sipser, algorithms and complexity with this book (not sure if it has a shorthand reference), compilers was the Appel book by preference, with several others acceptable.

Then, I would guess, someone complained about "difficulty". The administration got involved and things are more "approachable" now. Not so wonderful anymore.

Also, for me this book handled abstract algebra in much the same way that Sipser handles the theory of computation.