Topics in algebra

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1. seanhunterAug 2025
   There is no royal road to mathematics[1], and it's incredibly arrogant to think that any person can provide a single optimal path. For me for example the next steps are Axler, Abbott and Herstein[2]. That's where I am at the moment, and it's way earlier than the books listed here. It would be far from optimal for me to try to bang my head stubbornly on this list. Mathematics demands you put in the work to build a foundation - you cant just skip steps. For some people those books I listed are very rudimentary. For others they are definitely too advanced for where they are and they'll need something else.

   Even more so is the idea that you can actually cover the material listed in that page in 3 years. If you were to blast through it in that time you would only be skimming the very surface of the topics. There's simply no way you could possibly do all of those subjects justice in that time.

   [1] As Euclid is supposed to have said about geometry to the Pharoah Ptolemy when Ptolemy said he wanted to learn geometry but because of all the concerns of his kingdom he didn't have time to read the Elements.

   [2] "Linear Algebra done Right" by Sheldon Axler

   "Understanding Analysis" by Stephen Abbott

   "Topics in Algebra" by Herstein. this is a lovely book and beautifully written but some of the notation is a bit dated. I have two more recent algebra books but they are a bit advanced for me until I work through Herstein. They are Aluffi "Algebra Chapter 0" which is a good modern algebra book which introduces category theory at the start and Hien I forget the title but it's a springer one that he claims is good for an introduction but it's definitely not. It assumes you know a lot. It's very good though.

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2. asplakeAug 2025
   > "Topics in Algebra" by Herstein. this is a lovely book and beautifully written but some of the notation is a bit dated

   Now you make me feel old! I had the second edition just as it came out in 1984. A long time ago but I remember it as one of my favourites.

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3. graycatJun 2025
   For abstract algebra, there is the polished

   I.\ N.\ Herstein, {\it Topics in Algebra,\/}

   (markup for TeX word processing).

   For Galois theory, took an oral exam on what was in Herstein.

   For linear algebra where the field is any of the rationals, reals, complex, and finite fields there is

   Evar D.\ Nering, {\it Linear Algebra and Matrix Theory,\/} John Wiley and Sons, New York, 1964.\ \

   As I recall, Nering was an Artin student at Princeton.

   Some of the proofs for the rational, real, or complex fields don't work for finite fields so for those need special proofs.

   Had a course in error correcting codes -- it was applied linear algebra where the fields were finite.

   Linear algebra is usually about finite dimensional vector spaces with an inner product (some engineers say dot product), but the main ideas generalize to infinite dimensions and Hilbert and Banach spaces.

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4. graycatMay 2022
   Bluntly, with only occasional exceptions, have to learn from the books. Period. No biggie: Once are in grad school or a prof, have to learn from papers, and the good books are typically MUCH easier to read than the papers!

   For first calculus, read one of the well respected books. In my experience and opinion, don't need high school advanced placement calculus -- regard it as a waste of time. Also the last time I looked at Khan Academy, I concluded that it was a waste or worse and by people who didn't have a good background in calculus. Generally the Internet sources are too elementary. Solution: Just use some good books. Protter and Morrey is good (I taught from it), but it is a bit elementary. I learned from Johnson and Kiokmeister; I like it. At the time it was also being used at Harvard. The exercises are especially good. It's old, but calculus hasn't changed much since 1950 or so about when it gave up on teaching infinitesimals.

   But might check out the Maxima software for symbolic indefinite integration -- as I recall, its algorithms are so good they do any indefinite integration that can be done in closed form.

   For linear algebra, sure, Nering (a student of E. Artin at Princeton) and then Halmos (a finite dimensional introduction to Hilbert space and written when Halmos was an assistant to von Neumann, office not far from Einstein), etc. For still more, R. Bellman has a good book -- he went all over active and put in everything including the kitchen sink. Then for numerical work, old but a start, Forsythe and Moler and/or the documentation of LINPAK.

   For multivariate calculus through the divergence theorem and as needed, e.g., for Maxwell's equations, start with the most elementary treatments of vector analysis and f'get about the lack of proofs. Actually notice, can see via a Google search, that the divergence theorem on a box is just a dirt simple application of the fundamental theorem of freshman calculus. Then the difficulty is proving the thing on other shapes. Then do multivariate calculus again via, say, Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus. That he uses the Jordan curve theorem is a bit much but just go with the flow. Fleming's Functions of Several Variables does it with both measure theory and exterior algebra. Or W. Rudin's, as I recall, third edition, Principles of Mathematical Analysis. For calculus done via Lebesgue's measure theory, Royden's Real Analysis and/or the first half of Rudin's Real and Complex Analysis.

   My favorite author on ordinary differential equations is Coddington, and his elementary An Introduction to Ordinary Differential Equations is nicely done and likely enough for nearly any applications might encounter. Will want some of this if do A/C circuit theory or deterministic optimal control.

   For basic abstract algebra Herstein, Topics in Algebra should usually be enough and is well written. With that will be well prepared for, say, algebraic coding theory which actually has some important applications.

   Once have a background in measure theory, for probability, of course, Loeve or either of his students Breiman or Neveu. Breiman is really nice to read. Neveu is a good candidate for the most polished and elegantly done math book or writing of any kind in all of civilization. Be sure to get to the good stuff on the Radon-Nikodym theorem, conditional expectation, martingales, and ergodic processes.

   For relatively good results, better than hardly any students get just from courses, study the theorems and proofs until can do them yourself and can also explain them to any common man in the street and also explain where they fit in more generally. Also be able to do the exercises. E.g., Rudin's Principles has the inverse and implicit function theorems as exercises! Fleming applies them to Lagrange multipliers. These two theorems are the main prerequisites of differential geometry -- some parts of physics, now popular, touch on this math!

   In my Ph.D. program I led the class in 4 of the five qualifying exams, and for the analysis and linear algebra exams, the books above were the secret. After I did that, the department tried to have the students do better by offering a course from Rudin's Principles; the course didn't work; that book is too hard for a course and needs weeks of quiet time where a student can chew on the text slowly line by line. Rudin has some of the most precise presentations on Fourier theory. So when physics uses Fourier theory or engineers use the fast Fourier transform, you will likely know Fourier theory quite a lot better than they do. Right, the uncertainty principle in quantum mechanics is just a basic result in Fourier theory. With Fourier theory and linearity, Schroedinger's equation starts to make sense. A good exercise in early measure theory is differentiation under the integral sign, Leibniz's rule -- be sure to cover that with a good proof.

   With the background from the books above will be well prepared for a wide range of more narrow topics in pure and applied math.

   Nearly all these books are available used in nearly new condition for low prices. In a move I lost some of my library and rebuilt with used copies.

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5. he11owMar 2021
   There were a few milestone books, the kind that had such an outsized impact they spring to mind almost immediately:

   1. The Goal/ E. Goldratt. I read this first when I was a teenager, and kept returning to it (and some of its sequels) again and again. Adopting a systems thinking mindset. When analysing problems looking for the bottleneck, then figuring out how to elevate that constraint.

   2. Topics in Algebra/ I. N. Herstein. During the first summer at Uni I decided to read this book cover to cover and solve every single problem in it. Which I did. Sadly, twenty years on, my algebra game is poor. But that summer, of abiding in algebra, was a really spiritual experience. And it's left a mark.

   3. The Orchid Thief/Susan Orlean. By my early twenties it was clear to me I enjoy non-fiction more than fiction. But I enjoyed it for the intellectual experience, it would never leave me rattled the way fiction can leave you. And then I read this book and my mind was blown. I was outraged that people in the world can write like that, and that I am not one of those people.

   4. Destructive Emotions: A Scientific Dialogue with the Dalai Lama/D. Goleman. I randomly picked up this book, it was lying around at this place I was staying at. This was my first intro to Buddhist analytical analysis and I came out of it a different person. Not a better one, sadly, but for sure one more determined to learn more, and to bring these principles into my own life. It's been a journey since, but that's where it started.

   5. On the Phenomenon of Bullshit Jobs: A Work Rant/D. Graeber. It's kind of funny that this essay, which was tongue-in-cheek and deliberately provocative, would end up being like a real thing. But by the time I read it (and later, the full book) I was really questioning what was wrong with me that made everything feel so...meaningless. In his lingo, I had shit jobs and I had bullshit jobs, and I was really struggling to see how I can productively manage decades to come of work if these are the only two choices. But airing this problem in this way also helped me, eventually, walk down a path I was more comfortable with. It took a long time, but I finally like the path I'm treading.

   6. Not an essay, a film. Searching for Sugar Man/ M. Bendjelloul. I don't even know where to start on this one. I mean, the story's pretty radical as non-fiction goes (see above), but there were moments in and around it, where it's like someone comes and slaps you in the face to wake up. And then the meta aspect, of the director's own sad story. I spent A LOT of time thinking about this film.

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6. he11owMay 2020
   This. For Algebra, look for Topics in Algebra by Herstein. It reads beautifully, and the problems at the end of each chapters are very good at solidifying the knowledge.
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