I am an undergrad senior math major taking a gap year looking to become an actuary. However, I still want to continue learning pure math. I've been looking for a relatively high level text to self study for the next few months. I'm already a good chunk through the first chapter of Lang's "Algebra" and everything is flowing nicely. However, I have not done any of the exercises. Also, there are a lot of topological examples and things not purely algebraic which I would need to review (I have access to Munkres). In the end, I want to learn a lot of math (which could possibly help with my thesis) and solve a lot of problems. For background, my algebra class sophomore year used Artin's Algebra and I have also taken a seminar on algebraic combinatorics. Are there other texts (within or outside abstract algebra) better-suited for what I'm looking for? And finally, is the Companion to Lang's a supplement/fleshing out of the material, or more of a guide to getting at the solutions?
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1$\begingroup$ I used Lang as a student many years ago. My recollections of the book were that it was good but a little eccentric (e.g. look at the exercise on homological algebra). There some other choices, such as Dummit and Foote (sp?), that might work for you as well $\endgroup$– Donu ArapuraCommented Feb 5, 2023 at 4:12
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4$\begingroup$ @DonuArapura The more recent editions of Lang have an improved homological algebra section. $\endgroup$– Timothy ChowCommented Feb 5, 2023 at 4:39
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4$\begingroup$ did you look at Aluffi's Algebra 0 ? bookstore.ams.org/view?ProductCode=GSM/104 $\endgroup$– Dima PasechnikCommented Feb 5, 2023 at 9:31
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2$\begingroup$ Lang is much harder to read, and does not cover an increasingly important category theory point of view $\endgroup$– Dima PasechnikCommented Feb 5, 2023 at 9:33
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1$\begingroup$ I also recommend Aluffi's Algebra: Chapter 0. I solved pretty much every exercise when I was starting out in algebra, and though that's probably overkill, I don't feel like solving any on them had been a waste. $\endgroup$– Sergey GuminovCommented Feb 5, 2023 at 15:10
2 Answers
In my opinion, Lang's Algebra has an excellent choice of topics for someone who wants to do further work in algebraic number theory or algebraic geometry. I'm not sure whether I'd recommend it for self-study, however. My feeling is that the exposition, and the exercises, are rather uneven. If you're studying from it on your own, I think you could easily get stuck at various points where it's Lang's fault and not yours. But that's just my experience, and your mileage may vary.
Since you mentioned algebraic combinatorics, I feel obliged to mention Richard Stanley's Enumerative Combinatorics. This contains a ton of excellent exercises with estimated difficulty ratings and solutions, so I think it's perfect for self-study. Algebraic combinatorics is my field, so I'm biased, but I'd recommend Stanley over Lang for your purposes.
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$\begingroup$ In my seminar, I used Stanley and Fulton+Harris. So I do like and have experience with Stanley, but I do want to go further with RT and learn things like the Schur Weyl duality. Also, the standard text for commutative algebra at my school is by Atiyah and MacDonald and for Algebraic Geometry it is by Hartshorne. For algebraic number theory, I'd assume Lang is fine prep, but for algebraic geometry, would I be better off reading up on Atiyah MacDonald and maybe some of Artin or just use Lang (and the companion)? And finally, is Lang a good place to learn homological algebra? (new edition) $\endgroup$– Mr. OtisCommented Feb 7, 2023 at 1:09
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1$\begingroup$ You can't go wrong with Atiyah-Macdonald, which is a wonderful book (except there is no index to speak of), but I'd caution you against spreading yourself too thin. Given your goal, I think you are better off staying focused and working through one text, rather than dabbling in too many different texts. $\endgroup$ Commented Feb 7, 2023 at 1:50
In my opinion, lang's algebra is a not good place to learn homological algebra. I recommend Eisenbud's commutative algebra appendix 2 + Weibel's homological algebra.