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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.

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    $\begingroup$ 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. $\endgroup$
    – Qiaochu Yuan
    Oct 14, 2020 at 18:19
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    $\begingroup$ 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). $\endgroup$
    – Alekos Robotis
    Oct 14, 2020 at 18:23
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    $\begingroup$ 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. $\endgroup$
    – Justin Young
    Oct 14, 2020 at 19:23
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    $\begingroup$ 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. $\endgroup$
    – jons_stupid_questions
    Oct 14, 2020 at 20:12
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    $\begingroup$ 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. $\endgroup$
    – user507
    Oct 15, 2020 at 20:13

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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.

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    $\begingroup$ You can also accept this answer, which will prevent the question from rising to the top of the queue. $\endgroup$
    – Steven Gubkin
    Oct 19, 2020 at 12:57
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    $\begingroup$ 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.) $\endgroup$
    – KCd
    Oct 20, 2020 at 0:13
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    $\begingroup$ "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.) $\endgroup$
    – KCd
    Oct 20, 2020 at 0:14
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    $\begingroup$ evaluate it from the viewpoint of someone who does not already understand the content. My recommendation: look at Beachy & Blair's Abstract Algebra. $\endgroup$
    – KCd
    Oct 20, 2020 at 0:14
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    $\begingroup$ 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. $\endgroup$
    – jons_stupid_questions
    Oct 20, 2020 at 3:17

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