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Currently a 5th year PhD student, and I've been fighting tooth and nail to teach one of our junior-year honors sections in undergraduate algebra next fall (desperately hopeful we'll be able to return to normal in-person lectures at that point). Looks like it may come to fruition. Some questions on texts.

When I took undergraduate algebra, it was taught out of Herstein's classic Topics in Algebra for a two-semester honors sequence. The reception was somewhat lukewarm from the students. I definitely recall a point towards the end of the second semester (Galois theory) when about half the class was more-or-less completely lost, and from what I remember I was a bit lost myself but still managed to do well in the course. I liked the book at the time, but I remember feeling like it was times a bit slick, bit generally a good read overall and I learned a ton.

Recently, I've been made aware of another text by Dan Saracino titled Abstract Algebra: An Introduction, and apparently it's well-regarded by some people I respect. I've ordered a copy, but in the meantime I'm curious what others' opinions are on a comparison between the two, and experience with students' performance using both? In previous years, the course has used Herstein (and I believe at one point Artin, another great option but with a very different style).

The course is aimed at some pretty bright students, many of whom will pursue PhDs themselves, so I'm tempted to use Herstein (the problems are great, albeit at times very challenging, and it's one of the standards for this purpose). At the same time, I'm always of the opinion that students learn best when the exposition is clear and well-motivated, and it sounds like Saracino is a good candidate for that.

Any advice or anecdotes for people who have used both texts? I'm very passionate about teaching and I'd like to put the students in a good position for future graduate work. At the same time, I want the material to actually stick. I guess everyone has their own style, but I prefer courses that follow a text pretty closely rather than relying only on lecture notes.

For context, most students will be coming from a history of similar honors-type courses (including a recent course in Linear Algebra from Axler's excellent book, and an analysis course from baby Rudin, so they're well-versed in proof-writing and have a bit of maturity). It sounds like about half of students are expected to have taken a general intro to abstract algebra course (basic group theory, rings, and vector spaces) and half will be seeing it for the first time.

jons_stupid_questions
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    Qiaochu and Alekos - I think Artin is very valid recommendation here. I worked through the majority of the first edition when preparing for graduate algebra, and I think that prepared me for the material far better than Dummit and Foote (the other popular upper-level undergrad/beginning grad option). D&F has a weird way of giving so much context that you miss the point. I recently got a cheap copy of the 2nd edition, and it looks much improved (and thinner, a good thing since the 1st edition was a bit too comprehensive for that level). I'll have to think more about that option. –  Oct 14 '20 at 20:12
  • Hi Moo. Yes, I believe you're right. I'm a long-time user of the site, but a first-time poster. How do I request a moderator to move the topic? –  Oct 14 '20 at 18:08
  • Use the 'flag" link in your question and flag to a Moderator. –  Oct 14 '20 at 18:11
  • Thanks Moo, done. –  Oct 14 '20 at 18:14
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    I used Artin, taught by Artin (!), and I liked it; he was having us workshop the textbook for the next edition or something like that. I don't know anything about Herstein. People come to this site with a lot of questions from Dummit and Foote so I'm a little suspicious of it. There's some horrible textbook that teaches students that rings don't have multiplicative units which should be avoided for that reason alone. –  Oct 14 '20 at 18:19
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    I took a course a course from Goodman's book, which has the merit of being free. That book was alright, and probably not worse than average considering it is free. At the time it had some typos, though. I'm not a huge fan of Dummit and Foote, personally. I TA'd algebra from Artin and I think it was the best book for an "honors course" I've seen. Lang's Algebra is probably too hard to read at an undergraduate level, but in retrospect some parts are very clearly written. It definitely would not be a good main course text, though (at the undergraduate level). –  Oct 14 '20 at 18:23
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    Artin and Herstein are rather idiosyncratic. I don't like any algebra books, honestly, but the easiest one to read is probably Gallian. I also think Rotman is pretty approachable, though kind of boring like Hungerford or Lang. –  Oct 14 '20 at 19:23
  • Godement's "Algebra". – Jean Marie Becker Oct 14 '20 at 19:36
  • Huh. TIL Artin the textbook author is not the Artin I thought it was. In more relevant news, one might consider Aluffi's "Algebra: Chapter 0", though it is (imo) best suited for a second time seeing algebra. There's a lot of really great exercises, and it's as comprehensive as D+F without being D+F (though it emphasizes different things, obviously). – HallaSurvivor Oct 14 '20 at 22:53
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    Ah yes, assuming you're referring to Emil Artin. That would be his father, although Michael Artin is quite famous as well. As for Aluffi, I read a fair amount of it at one point and had mixed feelings... on the one hand, it's great that he introduces the categorical point-of-view from the beginning. On the other hand, I seem to remember not enough coverage on Galois theory? Also sits in a very weird place where it's not quite an undergrad book (definitely not a good first exposure to algebra), and not quite a grad-level book. – jons_stupid_questions Oct 15 '20 at 00:43
  • Nicholson's text is worth a look. There's a lot there. Also, Fraleigh is useful. – James S. Cook Oct 15 '20 at 04:07
  • Thanks James, I'll have a look. I've never understood why people complain about not being able to find a good algebra textbook... seems more like the opposite problem! – jons_stupid_questions Oct 15 '20 at 04:26
  • Right now, I'm really leaning towards Artin. I'll need to carve out the right syllabus, since it covers so much. But I plan to invite students to an "extra-credit" project due at the end of semester for motivated students to write an expository paper on a major theorem outside of the scope of the course (e.g., Hilbert's Nullstellensatz) along with useful applications, and I think Artin's perfect for that... – jons_stupid_questions Oct 15 '20 at 04:39
  • See also https://matheducators.stackexchange.com/q/2429/376, although the answers to that question cover / touch on only a handful of the many abstract algebra textbooks out there. – J W Oct 15 '20 at 09:48
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    Also, I suggest changing the title to one without the word "rant", even if you are using it tongue-in-cheek. – J W Oct 15 '20 at 10:02
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    Is this a one-year course? Personally, when I took this course as an undergrad it was a semester of group theory followed by a semester of rings and fields. First semester used Fraleigh, which I thought was fine. Can't remember the text for the second semester class, but I remember feeling that the motivation for Galois theory was incredibly weak. It seemed like some kind of archaeological artifact, like the highlight was supposed to be the insolubility of the quintic. I never had the faintest idea why anyone but a historian of science would care. –  Oct 15 '20 at 20:13
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    yep, one year. the plan is more-or less the same, but with a focus on getting to the Galois theory as quickly as we can in the second semester and really spend some time with it and go through it more slowly. I had a similar experience in undergrad. Very poorly-motivated introduction to Galois theory and it was left to the very end of the semester. Once you see Galois theory, it has a nice way of uniting most of the other topics covered in undergrad algebra, so I think it pays dividends to emphasize it and slow down the pace once we get there. – jons_stupid_questions Oct 15 '20 at 20:57
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    Perhaps, instead of editing your question to include the decision to use Lang, make it an answer to your question. – J W Oct 18 '20 at 10:14
  • By the way, I've modified the title in line with my comment above. You can always roll back the edit if you disagree. – J W Oct 18 '20 at 11:45
  • I've also added the (undergraduate-education) and (textbooks) tags to make your question easier to find. – J W Oct 18 '20 at 11:47
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    Thanks J W, done. – jons_stupid_questions Oct 18 '20 at 18:17
  • I used Saracino as an undergrad. The main drawback is that it doesn't have an answer key. Worked solutions would have really helped me. So that's what I'd wish for. – Benjamin Dickman Oct 18 '20 at 19:00

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Thanks for the advice everyone. After looking at many options and talking with faculty, it's been decided that we're using Lang's Undergraduate Algebra. I was leaning heavily towards Artin the past few days, but I was actually pleasantly surprised when I flipped through Lang's Undergraduate Algebra... far more standard organization, great exercises, and exposition is clear. It wasn't even on my radar (I was only familiar with the graduate version...) but this was strongly suggested by my advisor and it looks really fantastic.

jons_stupid_questions
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    You can also accept this answer, which will prevent the question from rising to the top of the queue. – Steven Gubkin Oct 19 '20 at 12:57
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    Let me warn you about something, based on my experience. I was the proofreader for the 2nd edition of the book and I too thought the book was fantastic. I recommended it to a postdoc at Harvard who was going to teach the undergrad algebra course there and didn't know what book to use; he was a recent PhD student of Lang but had not seen the undergrad algebra book before (only the more famous grad algebra book). He looked at it, like it, and decided to use it. At some point during the semester when he was teaching the course I asked him how it was going, and to my astonishment he said (contd.) – KCd Oct 20 '20 at 00:13
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    "They hate the book!" You see, the problem with judging a book when you already know the material is that you read it differently from someone who is seeing the material for the first time. The instructor and I saw all kinds of nice things in the book when we already understood the content. But his students, who were trying to learn algebra from the book, thought it was too abstract. And those were Harvard students. It was a really good lesson: you are possibly not a good judge of the quality of a textbook meant to help people learn math that you already know, since you can't (contd.) – KCd Oct 20 '20 at 00:14
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    evaluate it from the viewpoint of someone who does not already understand the content. My recommendation: look at Beachy & Blair's Abstract Algebra. – KCd Oct 20 '20 at 00:14
  • As another example of experts not necessarily being the best judges of a book, I know someone who was assigned to teach complex analysis at another Ivy League school and decided to use Stein and Shakarchi's complex analysis book for the course after looking through it and liking it. He found out at some point during the semester that the students did not like the textbook because the proofs were "too slick". That is, they could follow the proofs in it line by line but felt they were not able to learn methods of complex analysis based on those proofs to solve the homework problems. – KCd Oct 20 '20 at 00:22
  • KCd, the recommendation doesn't fall on deaf ears... that's something I'm acutely aware of since I was one of the students who was a "late bloomer" in undergrad. Most of the material wasn't sticking the first few years, and frankly I hated the Herstein/Rudin sequence I was taking. Sometime during my junior year, everything started to "click" and I started taking grad classes and doing really well. But I still very much remember the earlier years when I wasn't really getting much from my studies (or books). Nothing's been made official yet, so I may have another look at (cont.) – jons_stupid_questions Oct 20 '20 at 00:23
  • ...Beachy and Blair. I'm of the opinion that "hard" and "good" are NOT one and the same, and students are probably better off learning slightly less - but more deeply - than being confused and hating spending time with their book. But you're right, I looked at Lang's Undergrad Algebra book and thought it was an easier alternative to Herstein/Artin. That may very well not be true for the students. How is Beachy/Blair's coverage relative to Lang/Herstein/Artin? I don't own a copy. Any obvious weaknesses? – jons_stupid_questions Oct 20 '20 at 00:26
  • I'd say Beachy & Blair is a more user-friendly textbook than Herstein/Artin/Lang. It has a conventional coverage (unlike Artin) and includes a lot of examples. The price of the 3rd edition, which I've used a couple of times, was slightly over $50 at the time. I see the 4th edition, which came out after I last used Beachy & Blair, is now a bit over $60, so price creep is taking place. The last time I taught an undergraduate algebra course I used Thomas Judson's completely free online textbook: http://abstract.ups.edu/ and thought that was perfectly fine for the content I taught. – KCd Oct 20 '20 at 00:34
  • Can you explain how, as a graduate student, you are in a position to be considered as an instructor of an upper-level undergrad math course? The only time I've ever heard of a grad student teaching an upper-level undergrad math course (Galois theory), it happened because all the usual algebra faculty were going to be away the coming semester, so the professor in charge of assigning instructors was a bit desperate and asked a grad student if he was interested in teaching such a course. The grad student said "yes" but would never have asked to do this on his own initiative. – KCd Oct 20 '20 at 00:38
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    KCd - Since you asked, yes, I agree it's unusual. I'd rather not disclose too many details, but there was an issue with a faculty member leaving unexpectedly (and not on particularly good terms). I'm a fair bit older than the other grad students, and I formerly taught as an instructor (with a masters) at a larger public university for 5 years before joining a PhD program, so I at least have some track record teaching. Enrollment for this course will be pretty small, and there'll be quite a bit of oversight from faculty. FWIW, I plan to pursue teaching positions after, hence the enthusiasm. – jons_stupid_questions Oct 20 '20 at 03:17