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# Reddit mentions of Abstract Algebra, 3rd Edition

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Reddit mentions: 30

We found 30 Reddit mentions of Abstract Algebra, 3rd Edition. Here are the top ones.

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The answer is "virtually all of mathematics." :D

Although lots of math degrees are fairly linear, calculus is really the first big branch point for your learning. Broadly speaking, the three main pillars of contemporary mathematics are:

You might also think of these as the three main "mathematical mindsets" — mathematicians often talk about "thinking like an algebraist" and so on.

Calculus is the first tiny sliver of analysis and Spivak's

Calculusis IMO the best introduction to calculus-as-analysis out there. If you thought Spivak's textbook was amazing, well, that's bread-n-butter analysis. I always thought of Spivak as "one-dimensional analysis" rather than calculus.Spivak also introduces a bit of algebra, BTW. The first few chapters are really about abstract algebra and you might notice they feel very different from the latter chapters, especially after he introduces the least-upper-bound property. Spivak's "properties of numbers" (P1-P9) are actually the 9 axioms which define an algebraic object called a field. So if you thought those first few chapters were a lot of fun, well, that's algebra!

There isn't that much topology in Spivak, although I'm sure he hides some topology exercises throughout the book. Topology is sometimes called the study of "shape" and is where our most general notions of "continuous function" and "open set" live.

Here are my recommendations.

AnalysisIf you want to keep learning analysis, check out Introductory Real Analysis by Kolmogorov & Fomin, Principles of Mathematical Analysis by Rudin, and/or Advanced Calculus of Several Variables by Edwards.AlgebraIf you want to check out abstract algebra, check out Dummit & Foote's Abstract Algebra and/or Pinter's A Book of Abstract Algebra.TopologyThere's really only one thing to recommend here and that's Topology by Munkres.If you're a high-school student who has read through Spivak in your own, you should be fine with any of these books. These are exactly the books you'd get in a more advanced undergraduate mathematics degree.

I might also check out the Chicago undergraduate mathematics bibliography, which contains all my recommendations above and more. I disagree with their elementary/intermediate/advanced categorization in many cases, e.g., Rudin's Principles of Mathematical Analysis is categorized as "elementary" but it's only "elementary" if your idea of doing math is pursuing a PhD. Baby Rudin (as it's called) is to first-year graduate analysis as Spivak is to first-year undergraduate calculus — Rudin says as much right in the introduction.

Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.

Amazon search for Dover Books on mathematics

There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Pre-Calculus / Problem-SolvingCalculusLinear AlgebraDifferential EquationsNumber TheoryProof-WritingAnalysisComplex AnalysisFunctional AnalysisPartial Differential EquationsHigher-dimensional Calculus and Differential GeometryAbstract AlgebraGeometryTopologySet Theory and LogicCombinatorics / Discrete MathGraph TheoryP. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"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:

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:

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.

It should also be noted that simpler subjects and introductory texts tend to be common knowledge to the point that citation is often not needed. You don't need to cite that water is wet, not even on Wikipedia.

Journals and modern texts about modern subjects tend to be very well cited, because they're building heavily on other sources of information.

When you don't do this, you have to have an enormous amount of backgrounding. For instance, check out this algebra text. It's 944 pages because it doesn't tend to cite much of exposition and instead states it all directly. It includes an enormous amount of information -- it's meant to be used as fundamental education material. It's not just high level conclusions that could fit in 20-50 pages.

So the amount of citation depends a great deal on the purpose of the text and how close it is to common knowledge. However, Anita's criticism is clearly not common knowledge because nobody but her sees it the way she does. Therefore, she should be explaining how she comes to her conclusions, and citing information. She should also be citing the direct quotes she uses, because it's plagiarism otherwise (and we have huge volumes of evidence that she outright plagiarizes a great deal). Plagiarism in academia is something that ends your career.

Besides the Napkin Project I mentioned, which is a genuinely good resource? I got a coordinate-free treatment of linear algebra in my school's prelim. abstract algebra course. We used Dummit and Foote, which must be prescribed by law somewhere because I haven't yet seen a single department not use it. However, in reviewing abstract algebra I instead used Hungerford, which I definitely prefer for its brevity. But really, you can pick any graduate intro algebra text and it should teach this stuff.

You can try something like Artin or Dummit & Foote.

Harvard’s Benedict Gross gave a course using Artin’s textbook with lectures available on youtube.

It's hard to give an objective answer, because any sufficiently advanced book will be bound to not appeal to everyone.

You probably want Daddy Rudin for real analysis and Dummit & Foote for general abstract algebra.

Mac Lane for category theory, of course.

I think people would agree on Hartshorne as the algebraic geometry reference.

Spanier used to be the definitive algebraic topology reference. It's hard to actually use it as a reference because of the density and generality with which it's written.

Spivak for differential geometry.

Rotman is the group theory book for people who like group theory.

As a physics person, I must have a copy of Fulton & Harris.

I studied with this book on abstract. It's authoritative and brutal.

Learn Algebra, arguably the most important math subject (of course, I may be biased). Dummit and Foote is a fantastic intro if you have proof experience etc.

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 Proofscan 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.This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.

General Stuff:The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.

Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.

I know that in the long run competition math won't be relevant to graduate school, but I don't think it would hurt to acquire a broader background.

That said, are there any particular texts you would recommend? For Algebra, I've heard that Dummit and Foote and Artin are standard texts. For analysis, I've heard that Baby Rudin and also apparently the Stein-Shakarchi Princeton Lectures in Analysis series are standard texts.

I learned it out of Dummit and Foote originally, and I thought that was a pretty good book.

If you are asking for classics, in Algebra, for example, there are(different levels of difficulty):

Basic Algebra by Jacobson

Algebra by Lang

Algebra by MacLane/Birkhoff

Algebra by Herstein

Algebra by Artin

etc

But there are other books that are "essential" to modern readers:

Chapter 0 by Aluffi

Basic Algebra by Knapp

Algebra by Dummit/Foot

Well, Hardy & Wright is the classic book for elementary stuff. It has almost everything there is to know. There is also a nice book by Melvyn Nathanson called Elementary Methods in Number Theory which I really like and would probably be my first recommendation. Beyond that, you need to decide which flavour you like. Algebraic and analytic are the big branches.

For algebraic number theory you'll need a solid grounding in commutative algebra and Galois theory - say at the level of Dummit and Foote. Lang's book is pretty classic, but maybe a tough first read. I might try Number Fields by Marcus.

For analytic number theory, I think Davenport is the best option, although Montgomery and Vaughan is also popular.

Finally, Serre (who is often deemed the best math author ever) has the classic Course in Arithmetic which contains a bit of everything.

I'm soon starting my trek through every problem in the algebra text that Harvard's PhD prelim recommends for study:

Abstract Algebra by Dummit and Foote

I've started the first section of the first chapter, but that was only in a few hours of spare time. I'll be posting solutions by chapter soon and post my stories/insights on Hacker News. Here's section 1.1 (except the last problem, 36):

http://therobert.org/alg/1.1.pdf

Comments are appreciated. Better now than when I start the real journey. :)

http://www.amazon.com/Abstract-Algebra-David-S-Dummit/product-reviews/0471433349/ref=dp_top_cm_cr_acr_txt?ie=UTF8&amp;showViewpoints=1 is pretty popular and covers field theory.

Dummit and Foote's Abstract Algebra is an excellent book for the algebra side of things. It can be a little dense, but it's chock full of examples and is very thorough.

To help get through the first ten or so chapters, Charles Pinter's A Book of Abstract Algebra is an incredible resource. It does wonders for building up an intuition behind algebra.

If you are enjoying your Calc 3 book, I highly recommend reading Topology, which provides the foundations of analysis and calculus. Two other books I would highly recommend to you would be Abstract Algebra and Introduction to Algorithms, though I suspect you're well aware of the latter.

Introductory group theory normally comes from general introductions to abstract algebra. A few options ordered by level of difficulty (in my opinion) are

Gallian's book

This is a pretty gentle introduction. There are lots of exercises to help you master the subject and they are a good mix of easy and intermediate exercises. It takes its time which is why I think it's good for people without as much experience in proof.

Dummit and Foote

This is the most popular introduction, probably. It's well-written and it covers almost anything you might want to learn about algebra. It also has lots of exercises like Gallian, so if you go through and complete all the exercises, you'll know quite a bit.

Artin's Book

I haven't spent as much time working with this book, but I've heard it recommended a lot. It starts out with an overview of linear algebra and uses that as a motivator for most of the algebra and an example throughout the text. It's well-written too, but goes a bit faster than the first two.

I am a master's student with interests in algebraic geometry and number theory. And I have a good collection of textbooks on various topics in these two fields. Also, as part of my undergraduate curriculum, I learnt

abstract algebrafrom the books by Dummit-Foote, Hoffman-Kunze, Atiyah-MacDonald and James-Liebeck;analysisfrom the books by Bartle-Sherbert, Simmons, Conway, Bollobás and Stein-Shakarchi;topologyfrom the books by Munkres and Hatcher; anddiscrete mathematicsfrom the books by Brualdi and Clark-Holton. I also had basic courses indifferential geometryandmultivariable calculusbut no particular textbook was followed. (Please note that none of the above-mentioned textbooks was read from cover to cover).As you can see, I didn't learn much

geometryduring my past 4 years of undergraduate mathematics. In high school, I learnt a good amount of Euclidean geometry but after coming to university geometry appears very mystical to me. I keep hearing terms like hyperbolic/spherical geometry, projective geometry, differential geometry, Riemannian manifold etc. and have read general maths books on them, like the books by Hartshorne, Ueno-Shiga-Morita-Sunada and Thorpe.I will be grateful if you could suggest a series of books on geometry (like Stein-Shakarchi's Princeton Lectures in Analysis) or a book discussing various flavours of geometry (like Dummit-Foote for algbera). I am aware that Coxeter has written a series of textbooks in geometry, and I have read Geometry Revisited in high school (which I enjoyed). If these are the ideal textbooks, then where to start? Also, what about the geometry books by Hilbert?

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:

Undergraduate Algebraby Serge LangTopics in Algebra, 2nd edition, by I. N. HersteinAlgebra, 2nd edition, by Michael ArtinAlgebra: Chapter 0by Paolo AluffiAbstract Algebra, 3rd edition, by David S. Dummit and Richard M. FooteBasic Algebra Iand its sequelBasic Algebra II, both by Nathan JacobsonAlgebraby Thomas HungerfordAlgebraby Serge LangGood luck finding something useful!

If you can read through gallian's book, I consider dummit and foote's book (http://www.amazon.com/Abstract-Algebra-3rd-David-Dummit/dp/0471433349) as the best math textbook i've ever read. tons of examples, thorough treatment of material, and tons of exercises.

Friendly info:

"College Algebra" = Elementary Algebra.

College Level Algebra = Abstract Algebra.

Example: Undergrad Algebra book.

Example: Graduate Algebra book.

Yes: Dummit and Foote. I used it in my freshman algebra class. It has excellent proofs and exercises. It will teach you the mathematical maturity faster than analysis and will most likely be more useful to you later on.

Group theory is important for theoretical physics and crystallography, but I think it takes a back seat to the topics I listed. I've survived to grad school without learning anything beyond the basics, though I would love to study it eventually. Unfortunately, I couldn't fit in an abstract algebra course during my undergrad, so I don't have a textbook to personally recommend, although Dummit and Foote is popular with others.

Also, pretty much every branch of math (except maybe number theory?) is useful in physics (category theory, combinatorics, topology, measure and probability theory etc) so it's hard to make a comprehensive list.

Dummit (or just D&F), Artin, [Lang] (https://www.amazon.com/Algebra-Graduate-Texts-Mathematics-Serge/dp/038795385X), [Hungerford] (https://www.amazon.com/Algebra-Graduate-Texts-Mathematics-v/dp/0387905189). The first two are undergraduate texts and the next two are graduate texts, those are the ones I've used and seen recommended, although some people suggest [Pinter] (https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178) and Aluffi. Please don't actually buy these books, you won't be able to feed yourself. There are free versions online and in many university libraries. Some of these books can get quite dry at times though. Feel free to stop by /r/learnmath whenever you have specific questions

Griffiths' Quantum Mechanics has a crash course in most of the linear algebra required to do a first course in quantum mechanics. It's not very complicated - you just need basic understanding of vector spaces, linear transformations and functionals, and inner products, with a little bit of practice using dual notation of vectors (not too much, just enough for the Dirac notation which the book explains). Griffiths' also has a good explanation of simple fourier series/transform.

The key thing is being able to do basic linear algebra without matrices since in most of the cases, the vector space is infinite dimensional. But spin is a good example where almost everything can be done with matrices.

Additionally, solving ordinary differential equations and using separation of variables for partial differential equations in 3-d quantum mechanics would help.

Group theory will be of help in more advanced classes. Dummit and Foote or Arton's books on algebra are decent introduction. They are a bit dense though. If you want a real challenge, try Lang's Algebra book. I don't know of any easier books though. My first algebra book was Dummit and Foote which can be done without any real prerequisites beyond matrix algebra, but isn't really well written.

Links to books: Griffiths, Dummit and Foote.

PS: I have ebooks of these two books in particular.

If you want to do more math in the same flavor as Apostol, you could move up to analysis with Tao's book or Rudin. Topology's slightly similar and you could use Munkres, the classic book for the subject. There's also abstract algebra, which is not at all like analysis. For that, Dummit and Foote is the standard. Pinter's book is a more gentle alternative. I can't really recommend more books since I'm not that far into math myself, but the Chicago math bibliography is a good resource for finding math books.

Edit: I should also mention Evan Chen's Infinite Napkin. It's a very condensed, free book that includes a lot of the topics I've mentioned above.

Whoa, great questions, but I think you want a textbook, not a reddit post response. I used Dummit & Foote but it is probably a bit "heavier" than what you want/need at this point.