Abstract Algebra Textbooks
July 9th, 2007 by WaltIn comments, Grétar Amazeen asks:
Is Langs Algebra a good book? I just got it in the mail and I´m going to use it to brush up on my algebra before I start graduate school. I´ve heard that he uses his own private nomenclature, is that something I´ll have any problems with?
Michael has already given word-for-word my answer to the question:
Oh dear god no.
Lang does use his own private nomenclature (“entire rings”, for example), but that’s a minor issue. The book is just hard to read. The only chapter that I thought was well-written was the group theory chapter, but it’s very concise, so it might not be good for your purposes.
Abstract algebra has two excellent textbooks that are pitched at the advanced undergraduate or introductory graduate level: I. N. Herstein’s Topics in Algebra, and Michael Artin’s Algebra. Herstein covers the standard topics very clearly. Artin gives a much broader introduction to algebra’s relationship to other fields of mathematics, so it’s good for inspiration.
A few topics not covered in Herstein that are worth knowing are:
- The Nullstellensatz, and the relationship between algebraic varieties and ideals of commutative algebras.
- The theory of semisimple algebras, the Wedderburn-Artin Theorem, and its applications, such as Maschke’s Theorem for group representations.
(These are probably all covered in Artin, but I don’t have my copy handy so I’m not completely sure. They are all covered in Lang, but in both cases the chapters aren’t very good.)
A more idiosyncratic suggestion I have is Ideals, Varieties, and Algorithms, by Cox, Little, and O’Shea. It covers the Nullstellensatz, but from the point of view of Gröbner bases, which are a computational tool that makes it easy to work out examples in commutative algebra. They also make it easier to understand why homological algebra is interesting from an algebraic point of view, and not just as a tool in algebraic topology, again because they make examples easy to work out.
July 10th, 2007 at 4:19 am
Hæ Grétar! (replying in Icelandic since he is Icelandic)
langaði bara að kasta á þig kveðju. Ég var með Lang
á tímabili í einu námskeiði og hún er frekar þung og
mæli frekar með einhverjum öðrum bókum. Hvar ertu
að byrja í graduate school?
kær kveðja
Henning
July 10th, 2007 at 4:32 am
Thank you very much for this response.
I will have to check out Artin and Herstein. As for Cox, Little and O´shea, I read that one a few years back and didn´t like it. Some parts of it I thought where good, but I am no fan of the computational approach. I like my algebra a little more abstract and general.
Sæll Henning!
Ég er að byrja í Edinburgh. Er að fara í einhverja blöndu af algebru og rúmfræði, vonandi í óvíxlna algebrulega rúmfræði. Takk fyrir svarið með hann Lang. Ég þarf að skoða þetta. Gallinn er sá að ég hafði hugsað mér að komast eitthvað líka í víxlna algebru eftir Eisenbud og svo Hartshorne áður en ég byrja námið, veit ekki hvort ég hef tíma til að panta aðra bók og klára hana. Ertu á facebook?
kv, Grétarre
July 10th, 2007 at 9:35 am
I prefer Dummit and Foote or Artin as an introduction. Another very good book but slightly more advanced is P. M. Cohn’s “Basic Algebra: Groups, Rings, and Fields”. His “Further Algebra and Applications” is quite remarkable IMO. I’ve never gotten the chance to read his Intro version “Classic Algebra” but I suspect it could be good as well.
July 10th, 2007 at 11:48 am
birkhoff & mac lane is good.
but mac lane & birkhoff is even better.
i’ve definitely gotta look into this
cox, little, & o’shea you speak of.
“oh dear god no” is of course
the correct response as to lang.
July 10th, 2007 at 2:03 pm
Lang is pitched at a different level than Herstein and Artin, and is more comparable to say, Jacobson. Also, neither of Herstein or Artin have anything on noncommutative rings.
July 10th, 2007 at 10:56 pm
I, for one, happen to like the Lang algebra book quite a lot (as with most of his books that I know). He has a very direct, no-nonsense style and gets right to the point. He tends toward abstraction (which in my view is a plus), and illustrates the relationship of algebra to other branches of mathematics. I like the term “entire ring” and other coinages of his (e.g. “toplinear isomorphism”). Finally, the problems are mostly interesting; there are few routine exercises (though of course this is a matter of opinion).
If Lang has a flaw, it is that he can at times be somewhat more ad hoc than I would prefer. (I despise ad hoc mathematics.) To get an idea of where I am coming from, I’ll reveal that my favorite text on algebra (and most core topics) is Bourbaki. (I can already hear the shouts of incredulity…)
July 12th, 2007 at 12:42 pm
I don’t know if I’d call Artin a “graduate” textbook. I used it as an undergrad, and I know of plenty of cases where it’s used as an undergrad text but none where it’s used as a grad text.
It’s definitely a lot more readable than some of the graduate texts, though.
Lang isn’t a bad textbook, but I get the feeling that he tried to cover too much and as a result doesn’t cover anything in depth. I have the sense that it’s more useful as an encyclopedic reference than a textbook, although having not done any work with algebra since the course I took that nominally used Lang as a text, I don’t really want to stand behind that statement.
July 12th, 2007 at 12:55 pm
“[Lang is] more useful as an encyclopedic reference than a textbook”
Exactly.
July 12th, 2007 at 2:26 pm
Herstein and Artin should be fine if the goal is to review undergraduate material in preparation for grad. school. For the material on semisimple algebras, and noncommutative algebra in general, I think Jacobson (Basic Algebra) is quite readable.
July 12th, 2007 at 5:42 pm
By the way, the Nullstellensatz is definitely covered in Artin; I don’t think the things listed in (2) are, although I don’t have my copy at hand so I cannot be totally sure. (Note that the Artin of the algebra text is the son of the Artin of the Artin-Wedderburn theorem.)
July 12th, 2007 at 7:01 pm
I don’t know the Artin book very well, but I seem to recall that he begins with matrices and determinants, which, for a committed Axlerian like me, is a huge no-no. (Obviously determinants are important and should be discussed, but not at the beginning, for heaven’s sake!)
“[Lang is] more useful as an encyclopedic reference than a textbook”
At the graduate level, there shouldn’t be a distinction.
July 13th, 2007 at 4:58 am
James, I really have to disagree. Are you seriously suggesting that graduate students are suddenly imbued with some magical ability to absorb information, and that there’s no longer such a thing as good and bad pedagogy?
July 13th, 2007 at 2:47 pm
>John Armstrong Says:
>> “[Lang is] more useful as an encyclopedic reference than a textbook”
>Exactly.
Exactly wrong. Encyclopedic are useful because you can find things in them. To do that you need to know what they are called.
July 13th, 2007 at 2:55 pm
Doh…Encyclopedias.
Look, for those of you who think I am being too harsh here, I would defend myself by saying that I am sure that Lang’s Algebra book is an excellent exemplar of something, but a graduate textbook of - or encyclopedic reference for - Algebra, are not it.
July 13th, 2007 at 3:27 pm
I find Lang’s terms are close enough to find what I need. My point is that it’s really an awful text to try to learn anything from, since he does things in a way that will confuse the hell out of you if you don’t already know what his point is.
That Lang effect is somewhat ameliorated when he’s there to tell you what he means. But he’ll do it by screaming at you.
July 13th, 2007 at 7:41 pm
“James, I really have to disagree. Are you seriously suggesting that graduate students are suddenly imbued with some magical ability to absorb information, and that there’s no longer such a thing as good and bad pedagogy?”
I’m not quite suggesting that. Rather, what I’m suggesting is that the meaning of “good pedagogy” depends on the level of the student. In graduate school, the student should be making the transition to mathematical adulthood, where “good pedagogy” means simply “good exposition”–which, in turn, means a presentation of the facts in as complete and clear a manner as possible. One should be viewing the subject in its full Platonic glory, so to speak. (This is presumably the same thing that one aims for in an “encyclopedic reference”–at least it should be.)
For whatever reason, some students, at earlier stages in their careers, are simply not confortable enough with the level of abstraction that is required for this type of presentation, so their textbooks have to talk down to them to some extent. Speaking for myself, however, I have always resented this kind of condescension, which consists mainly of two procedures: (1) omitting important parts of the story that the author considers to be “too hard” (as in “the proof requires advanced concepts and will not be given here”); and (2) spreading information out over as much page space as possible, making it virtually impossible to tell at a glance what the key idea is, apparently in the hope that this will make the key idea easier to digest. Actually, now that I think about it, there is yet a third (related) technique for writing “undergraduate” level texts, which is to present topics out of their logical order.
Some people, I suspect, mean exactly this type of patronizing style when they talk about “good pedagogy”. There may even be students for whom it actually is good pedagogy. For me, however, pedagogy of this sort is–at best–a necessary crutch, to be dispensed with as soon as “mathematical maturity” permits. In my own case, I never wanted it to begin with; I would have preferred Bourbaki in middle school.
Naturally, I would be curious to hear from those with different preferences.
July 13th, 2007 at 8:44 pm
One should be viewing the subject in its full Platonic glory, so to speak. (This is presumably the same thing that one aims for in an “encyclopedic reference”–at least it should be.)
I’d say no on both counts. I’m still learning new ways to see the basic structure of a group, and of the general theory of groups (just to name an example) over fifteen years after I first ran into them. Should a graduate algebra textbook be written in the language of Lawvere’s “functorial semantics”? Hardly.
There are still certain things, even at the graduate level, that are simply far more abstract than are called for. An analyst will need to know that a group is a set with a composition. An algebraic topologist would benefit from knowing a group is a category with one object. A quantum topologist should know that a group is a group object in the category of sets. No general text needs to go down to that level.
July 14th, 2007 at 7:27 am
There are still certain things, even at the graduate level, that are simply far more abstract than are called for. An analyst will need to know that a group is a set with a composition. An algebraic topologist would benefit from knowing a group is a category with one object. A quantum topologist should know that a group is a group object in the category of sets. No general text needs to go down to that level.
How can you be so sure of this? Indeed, I would be willing to bet that, in a hundred years’ time, basic texts will in fact be written at such a level and nobody will think anything of it. Any abstraction that is illuminating or useful eventually becomes mundane mathematics, no matter how esoteric it may have seemed when first introduced.
So how can we be certain that an analyst would not benefit from exposure to the points of view that you mention? (And, by the way, why should we assume that no one called an “analyst” would ever simply find those points of view interesting for their own sake?) Of course, the very notion that mathematics should be presented as “facts you will need to know to do your job” is wrongheaded (not to mention distasteful) in my view. It’s wrongheaded in Calculus I, and it’s wrongheaded in mathematics graduate school.
And let’s remember that we’re not talking about Lawvere’s “functorial semantics” here. We’re talking about Lang’s Algebra, where, for example, a group is defined as a monoid with inverses—quite suitably for your “analysts”. So where’s the problem, exactly?
July 14th, 2007 at 3:00 pm
Would that in a hundred years’ time we were teaching everyone about group objects and so on. However, despite the idealism that comes with being a mathematician I’m pragmatist enough to recognize when people just aren’t getting it. And people just don’t get internalization right off as it stands now.
You have to keep in mind that Serge was very much not writing his texts in a vacuum. He is not Bourbaki. Serge was very much interested in keeping touch with social context, either in society at large or within the cloistered society of academic mathematics.
He tried to split the difference between generality and teachability — especially what was teachable back in the late 1960s at Columbia and then the late 1970s at Yale. Is his result perfect? Far from it. But as it stands it is not suitable as a text for the first-year gradute course in algebra, either at the level of Yale or at the level of the University of Maryland. In the current context it’s eminently suitable as a general reference. If I need a quick refresher of a basic fact about fields it’s the first place I look. If I need something more detailed about a more specific field, there are other canonical references I’ll use for them.
July 14th, 2007 at 3:03 pm
And I brought up Lawvere because you said that one should view the subject in its “full Platonic glory”. The category $latex \mathrm{Th}(\mathbf{Grp})$ is exactly that for groups, but (wisely) nobody teaches in those terms in a first graduate course in algebra.
July 14th, 2007 at 6:37 pm
Obviously, what constitutes “full Platonic glory” changes over time; the subject progresses and becomes ever more abstract. But the point is that one should present the “full Platonic glory” as seen at some point in history. The contrast I’m making isn’t between two definitions of group, one more abstract than the other. The contrast I’m making is between two expository procedures: one that consists in presenting the definition of a group upfront, clearly, distinctly, and abstractly; and another that may, for instance, involve presenting a verbose discourse on a particular example of a group before giving the definition. (Note that in both cases, I assume that it is the general concept of a group that is of interest; but in the second case, the author is reluctant to explicitly discuss the general object without giving “preparation” that involves previously familiar objects.)
It is because Lang follows the first procedure that I regard his book as appropriate for a graduate course (as, by the way, did Lang himself as of 2004). Now, you say that it is not appropriate for this purpose; I would like to know on what you base this opinion. Let’s get specific: what would be an example of an unsuitably written passage in Algebra, and how would you change it to make it more intelligible?
July 15th, 2007 at 11:37 pm
I read both Artin’s and Herstein’s book in my undergrad days and I admit that I loved Artin disliked(read hated!!!) Herstein. P.M Cohn’s book is good as it compliments Artin.
About Dummit and Foote I will just like to say that it is a good reference in terms of all theorems and all but not at all a good book to read.
In a nutshell I think Artin is the best introductory level book (may be complimented with Cohn), and if you are impressed by geometric arguments then probably Artin is a must.
July 16th, 2007 at 1:46 am
John,
I think you’ll find that Serge was in fact a Bourbaki, at least that seems to be the general consensus (see http://en.wikipedia.org/wiki/Bourbaki for instance). This doesn’t take away from your point that Serge had his own style. I find Lang’s Algebra a very challenging and useful book. But I don’t think its good as a first graduate text on the subject. I’ve found this to be quite an interesting topic, thanks Walt!
July 16th, 2007 at 2:27 am
Shane, Serge was a participant, but he broke ranks with them in writing the bulk of his texts. The style is radically different, as he would be the first to tell you (very loudly).
July 16th, 2007 at 3:30 pm
I agree that Lang’s style is distinct from Bourbaki’s, but I think to say they are radically different is an exaggeration. There are books that are much more un-Bourbaki than any of Lang’s, both in style and content. This may perhaps be easier to see in an area like analysis than it is in algebra: Lang’s treatment of analysis (and, especially, differential geometry) very consciously reflects the influence of Dieudonné, who is about as Bourbaki as it gets.
July 17th, 2007 at 5:44 am
for the more computationally minded (like me), i thought yaps book “fundamental problems in algorithmic algebra” was (and still is) excellent. he has a preprint of the book online for free academic use:
http://cs.nyu.edu/~yap/book/alge/ftpSite/
from my perspective, this book answered all the questions about computational algebra i didn’t even know how to ask.
July 18th, 2007 at 2:46 pm
I haven’t looked at Lang’s book, but I used a mix of Dummit & Foote and Herstein back in college. Personally, I preferred Herstein between the two, but find the other a good reference now.
July 22nd, 2007 at 9:33 pm
I’ve never posted here, but I’m an avid reader and I’d thought I’d chime in. I’m a pure math student at MIT, where I studied algebra under Artin himself — with his book, of course. I found Artin’s text distasteful for the first few months in the course, owing to (what I perceived to be) its sloppy and impromptu-like motivation of major concepts. I have a better opinion of the book today, but it’s hardly approaches what I imagine a good math text to be.
This brings me to this divisive claim:
<i>The contrast I’m making is between two expository procedures: one that consists in presenting the definition of a group upfront, clearly, distinctly, and abstractly; and another that may, for instance, involve presenting a verbose discourse on a particular example of a group before giving the definition. (Note that in both cases, I assume that it is the general concept of a group that is of interest; but in the second case, the author is reluctant to explicitly discuss the general object without giving “preparation” that involves previously familiar objects.)</i>
The distance between these expository procedures and the pedagogical techniques they imply (which, it should be said, are not a true bifurcation; there’s clearly a gray area in between) divide mathematicians into two groups: those who enshrine the importance of motivating mathematical ideas, and those who have already intuited that motivation and are prepared to work with abstraction. As a pure math student, I of course value mathematics for the sake of mathematics; but one thing I hate and loudly decry is the idea that mathematics (say even at the graduate level) should be taught upfront — before any motivation, before any discourse about its development and the motivation of major ideas — in full-fledged abstraction. Mathematics is never envisaged or developed this way and should certainly never be presented this way to students. There’s a reason we call graduate students graduate students and not professional mathematicians.
That said, I share James’s disgust for those authors who obscure the big picture by disingenuously leaving parts out. One of the most helpful tools in understanding a major concept is to understand its limits or its difficulties (counterexamples, for example — no pun intended). Although I admit one cannot always give a thorough treatment of a subject, one should never forget to include in the exposition of a subject hints about its messier details or its location in the big picture.
To demonstrate the kind of mathematical exposition I consider ideal, I cite Munkres’s <i>Analysis on Manifolds</i> and John Lee’s <i>An Introduction to Smooth Manifolds</i> (from which it’s not hard to guess that my main interests lie in differential geometry and mathematical physics).
In the end, I believe this idea of working from a totally unmotivated, abstract definition of mathematical structures belies the human work that actually leads to its development — and stems from a tacit arrogance. Every mathematician — and even us lowly, helpless students — will admit the beauty in our work lies in its pure, Platonically ideal abstract structure. Only bad mathematicians will suggest they’ve never needed the more worldly concrete examples to help them better understand the implications of that structure.
July 24th, 2007 at 9:14 am
matt:
I do not disagree with you that “motivation” is important. What I would point out is that familiarity is not the only, nor necessarily the most important, source of motivation. We should not forget that simplicity is also a motivation, and for me it is a much more important kind. For example, the “motivation” behind the axioms for a topological space is the fact that there are only two of them (invoking nothing but elementary set theory), which nevertheless suffice to imply a whole host of “spatial” properties. Thus, for me, the ideal books on general topology–for all purposes, including a first course–are those of Kelley and Bourbaki
A certain type of person would complain about this type of presentation, on the grounds that the simple set-theoretic axioms seem “unmotivated”, having apparently nothing to do with space and continuity. Such a person, in my view, would be missing the whole point of the subject. Of course it’s “unintuitive” (at first) that such a theory could be developed from such premises; but the most important objective of mathematical activity is to expand and develop our intution–not to codify the limitations of our imagination. The axioms for topology do not exist to summarize our pre-existing spatial intuition; they exist in order to allow us a glimpse of the deep relationships between our mundane universe and more exotic worlds. In this sense, the very beauty of mathematics resides in its (initial) “unintuitiveness”.
July 24th, 2007 at 2:41 pm
James –
The axioms for topology do not exist to summarize our pre-existing spatial intuition; they exist in order to allow us a glimpse of the deep relationships between our mundane universe and more exotic worlds. In this sense, the very beauty of mathematics resides in its (initial) “unintuitiveness”.
Of course I agree — this is exactly what I meant in my penultimate sentence. But pedagogically it makes no sense to approach this level of generality without first mastering the intuition that led to its study and formulation in the first place. No student — graduate or undergraduate — initiates a study of cohomology theories using the Eilenberg-Steenrod axioms before actually understanding more explicit instantiations of cohomology theories (de Rham, before anything else, usually) and how they can be related to make those axioms sensible. Neither should he or she read an introductory gauge theory text that fails to trace (even if briefly or only heuristically) the field’s development from fundamental insights about the manipulation of Maxwell’s equations. The axioms for topology indeed do not “exist to summarize our pre-existing spatial intuition”; but they exist precisely because someone first had to conceive of the right axioms — and it seems to me of utmost importance for students to learn how the greatest and most helpful abstractions have developed from messier, less general details. While the beauty of mathematics may consist in the elegant simplicity of its abstractions, it might as well be defunct if its practitioners don’t understand how to build new structures or develop new insights — in your words, to “glimpse the deep relationships between our mundane universe and more exotic worlds.” The power and vitality of mathematics resides in that very ability to “expand and develop.” Except for the most gifted and perspicacious among us, the cultivation of that ability usually requires learning about how others have formalized the great advances in the past.
Ultimately, I think we only disagree in what we understand to be the level of mathematical maturity of an average mathematics graduate student. I personally find it highly unlikely that most first- or even second-year PhD students have so thoroughly mastered a subject that they need, or even desire, a text like Lang’s — unless of course that subject happens to be one’s primary field of interest. But then again, there’s nothing incredibly hard about Lang’s text (in the sense of being beyond the abilities of any graduate student of pure mathematics) — except that it’s so dry and often unenjoyable to read. Perhaps the issue always devolves to one of taste, as it always has for me with the case of Spivak’s Calculus on Manifolds. Many of my peers exalt that compact little text; I’d rather have Munkres — no matter how much more verbose he may be — any day.reCAPTCHA WP Error:incorrect-captcha-sol
July 27th, 2007 at 9:41 am
matt:
You have essentially made an argument for studying the historical development of a subject alongside the subject itself. I have no objection to doing this, and indeed, I try to do it myself as much as I can (the beautifully written historical notes are one of the things I love about the Bourbaki volumes). There’s no doubt that knowing the history of a concept can often help in developing a deeper understanding of the concept.
But that doesn’t undermine my contention that simplicity is to be valued more highly than familiarity. And note that the history of some subjects, such as quantum mechanics , can actually be a hindrance to understanding.
You’re right, I think, that our disagreement is about how much mathematical maturity to expect from students. I would simply argue that, regardless of the level they arrive with, they should be expected to reach the level of Lang (say) as quickly as possible.
It may be true, as a matter of fact, that no one actually begins the study of cohomology with Eilenberg-Steenrod, but perhaps they should. I certainly wouldn’t want to dismiss the idea out of hand. However, I’ll have to think about that before coming to any firm conclusion. The same goes for gauge theory–remember the example of quantum mechanics, after all.