Picking the right Abstract Algebra introduction book for the purpose of learning Cryptocurrencies(Zero-Knowledge Proofs and Cryptography)

Question

I know that a lot of people have asked for book recommendations on Abstract Algebra before, but I hope that this thread is different because of the predefined list and specific use-case.

I've started with Pinter, since it was readily available, but it feels like either - things are not explained in enough depth, ie. something is mentioned without the examples needed for me to understand in on a deep level - ex. group acting on a set or too many exercises that ask you to do the same thing, ie. it gets super boring - for example most exercises in the Cyclic Groups/Order of Group Elements chapters. The chapter on Permutations of a Finite set was satisfying. Another issue I have is that most of the time the examples are too simple, so I feel like I want to solve the generalized exercise right away.

Now my reason for wanting to learn Abstract Algebra is because I want to understand Cryptography on a deeper level and also to be able to understand Cryptocurrencies which in the recent years have started to use a lot of complicated Math - ie. ZK-Starks, Smart Contracts, etc. I am also interested in Homomorphic Cryptography(encrypted, read untrusted, computation).

At first I started with some Number Theory, but I got put off at the Quadratic Reciprocity proof, it seemed too geometrical. When I started looking for another proofs, they were all algebraic, ie. I found out that I was learning Elementary Number Theory, while Algebraic Number Theory seems to be where the cool proofs are at. So then I found out I don't really know enough Algebra - I did study a bit of Rings/Fields/Linear Algebra/Topology/Analysis/Set Theory/Number theory, but I never learned about Groups, Field Extensions, Galois Groups, Homomorphisms/Isomorphisms, Quotient groups/rings, etc.

I've taken the time to gather a list with some basic information about all of the recommended books in similar threads:

1. I. Herstein - Abstract Algebra, 1996, 3rd edition, 240 pages
2. G. Lee - Abstract Algebra, An Introductory Course, ..., ..., 250 pages
3. D. Saracino - Abstract Algebra - A First Course, 2008, 2th edition, 279 pages
4. J. Rose - A Course on Group Theory, 2012, n/a, 300 pages
5. C. Pinter - A Book of Abstract Algebra, 2010, 2nd edition, 340 pages
6. T. Judson - Abstract Algebra, Theory and Applications, 2019, annual edition, 325 pages
7. J. Fraleigh and V. Katz - A first course in Abstract Algebra, 2003, 7th edition, 470 pages
8. T. Hungerford - Algebra, 2003, springer edition, 485 pages
9. T. Hungerford - Abstract Algebra, An Introduction, 2014, 3rd edition, 500 pages
10. M. Artin - Algebra, 2010, 2th edition, 513 pages
11. J. Gallian - Contemporary Abstract Algebra, 2017, 9th edition, 547 pages
12. J. Rotman - A First Course in Abstract Algebra with Applications, 3rd edition, 580 pages
13. A. Knapp - Basic Algebra, 2016, 2nd edition, 590 pages
14. D. Malik and J. Mordeson and M. Sen - Fundamentals of Abstract Algebra, 1996, n/a, 600 pages
15. S. Lang - Algebra, 2002, 3rd edition, 867 pages
16. D. Dummit and R. Foote - Abstract Algebra, 2003, 3rd edition, 892 pages

• Dummit and Foote, Lang - seems too be too long for my purposes
• Hungerford - Algebra - I liked the intro chapter, after that I guess it feels too rushed
• Hungerford - Abstract Algebra, An Introduction - I prefer for groups to be introduced before Rings
• Pinter - I've explained above, perhaps I should continue with it

It would be great if someone could comment on the rest of the books, or even to add another book to the list or perhaps just point me in the right direction.

Answer 1

I don't have enough points to comment so I'll answer this question. From your question it looks like you're about to self-teach in which case you want a book that spells out every step(you don't have an instructor to ask a question about every triviality), has crapillion examples(to discern general patterns of how algebraists think; not a priority in books geared towards classroom use) and ain't worried about students cheating(for example, some books won't provide answer key because ... cheating; you're self-motivated and so that's not your problem). This rules out most classics, especially those heavily used in classrooms.

I am familiar with every one of these books because I had to self-teach myself algebra. Out of all the books you listed, #14 comes closest to satisfying the bullets I listed above. In general, Indian authors are well-known for this style of writing. For example, consider the following book: A Course in Abstract Algebra, 5th Edition By Khanna V.K. & Bhamri S.K. Consider the level of detail in this book. It's insane for a college level textbook.

In support of what I said about Indian math textbook style, here take a look at other boooks on other topics:

Real Analysis by Bali

Galois Theory by Rajnikant Sinha

Krishna's Topology by Sharma

If I recall correctly, Knapp, Rotman, Lang and Hungerford have two books each on abstract algebra. One introductory and the other - grad level. Out of these four, Hungerford provides more examples and more detail. Lang has very idiosyncratic style that takes some getting used to (I think all his books are like that). Another book in the list that gives off slightly "weird" vibes is the one by Artin. Speaking of grad level books, you forgot other "superstars" such as Jacobson and Aluffi :) The books by Gallian, Fraleigh, Judson, Herstein are arguably the easiest ones to read. One thing about your list, though, is it's missing a lot of newer books and there are a ton. For example, if you like pretty colors and/or nice fonts and/or hand-holding, consider the following:

Intro To Group Theory by Nash

Abstract Algebra by Warner

Abstract Algebra by Dos Reis

Abstract Algebra by Terras

Abstract Algebra by Jackson

There are also a lot of older books that are massively easy to read but not talked about much. For example,

Numbers and Symmetry by Johnston/Richman. This book is pre-abstract-algebra of sorts. There exists pre-calculus. Why not pre-abstract-algebra?

The fascination of groups by Budden. A bona-fide textbook on groups. Just not written/structured like your regular, vapid book that comes out of a typical textbook mill.

Rings and Factorisation by David Sharpe

Long story short, you don't need to know a whole bunch of algebra to study the topics of your main goal. You just need to know a few fundamental topics really well to the point that (1) you don't get scared and run away when you come across terms like "magma" and "homomorphism"(rather have the maturity to look up their definition and a few examples) (2) have the technical dexterity(dirty calculations using pen and paper) in just a few fundamental topics. University textbooks (usually) are not designed for that. They are usually heavily padded with extraneous info and the topics/pedagogical style is chosen because of traditions and bureaucratic authority.