Book Note: A little light algebra

Aluffi algebra Suppose, e.g. as a philosopher of mathematics, you want to get to know more mathematics (after all, it is always a jolly good idea for a philosopher of X to know more than a mere smidgin about X). One area which you have seen to be absolutely central to the modern mathematical curriculum is abstract algebra. Fine: but what to read if — quite a big ‘if”! — you do decide to get to know more about this?

It depends what base you are starting from, of course. At a really introductory level (first year undergraduate, perhaps), one really nice option which I can warmly recommend is Alan Beardon’s relatively short Algebra and Geometry (CUP, 2005), which is very well put together and indeed not-too-abstract. But perhaps this doesn’t take you far enough to get a more rounded sense of modern algebra.

So suppose you want more than you’ll get from Beardon; or perhaps you already have some — possibly fragmentary, possibly half-remembered — knowledge of algebra, and want to go up a level in sophistication and detail. What then? One option is Saunders Mac Lane and Garrett Birkhoff’s Algebra (3rd edition, AMS Chelsea, 1999). This is a distant and slightly more advanced descendant of their famous A Survey of Modern Algebra (which originally dates back to 1941), and is notable for the way it weaves into the story categorial ideas, with a fairly light hand and in an illuminating way. But the treatment is quite brisk, and I think there is now a better option, taking quite a similar approach, but in a rather more engaging way. This is Paolo Aluffi’s Algebra: Chapter 0 (American Mathematical Society, 2009). Its presentation of even quite complex ideas is typically exemplary, both in giving motivation, and in explaining official definitions, presenting the proofs etc, and all often written with a light touch. So this is admirable. and I think is particularly suitable for self-study.

The chapter titles indicate the coverage. I, Preliminaries: sets and categories (so yes there is early introduction to categorial ideas, but again done with a light touch, emphasising the notion of ‘universal properties’). II, Groups, first encounter (the basics done very clearly). III, Rings and Modules (getting as far as a first look at complexes and homology and exact sequences). IV, Groups, second encounter. V, Irreducibility and factorization in integral domains. VI, Linear algebra. VII, Fields. VIII, Linear algebra, reprise. IX, Homological algebra. So things eventually get pretty serious, and you can bail out well before the end but still with a good sense of the topics and approach of modern algebra.

These days, alternative standard recommendations for tackling algebra at this sort of level include e.g. Serge Lang’s Algebra (3rd revised edition, Springer, 2002), and David Dummit and Richard Foote’s Abstract Algebra (3rd edition, Wiley, 2004). But these are quite a bit longer than even Aluffi’s weighty volume, and really cover unnecessarily much (for our purposes). And, more to the current point, I would say neither is a particularly attractive read.

So headline summary: if you want to learn some algebra, dive into Aluffi’s Algebra: Chapter 0.

9 thoughts on “Book Note: A little light algebra”

  1. Unfortunately, it’s very expensive. Amazon UK’s price is £74.95, and the lowest price offered there is £65.31. In a way that’s understandable, since at 713 pages it’s not hugely out of line with the prices of some other maths books. For example, Visual Group Theory, another book in the same AMS series as Chapter 0, is £50 for 306 pages.

    But why do any of them have to be that expensive? Allen Hatcher’s Algebraic Topology, 556 pages, from a different publisher, is only £26 and is also available free online. Indeed, there are many examples of more reasonably priced books across a wide range of subjects.

    I wonder how it works from an author’s point of view. Do they get more money from sales of expensive maths books? Normal economics is suspended, to some extent, for many maths texts, since university libraries will buy a copy and since it might be picked for university courses, so that students have to buy it. Otherwise, I’d have thought that authors might do better from less expensive books (or else no worse), by selling more copies.

    1. Sadly, it is indeed very expensive (though, as you say, not out of line for books of that size). And yes, I do think that authors should choose a publisher who either goes for simultaneous paperback printing, or allows you to keep a late version of the book on your website, or allows you to make the book freely available after a certain interval (or some combination). CUP, for example, is increasingly good about this.

      However, I feel reasonably free to ignore price considerations in the various book recommendations I make. Officially, above the line, I’m assuming that most of those likely to be wanting to follow up any book recommendations here will have access to a university library (and Aluffi’s book should be in any half-decent one).

      And unofficially, below the line, the fact is that (almost?) everyone knows how to get PDF or DJVU copies of (almost?) anything I mention. Not that we condone such wicked behaviour, of course.

      1. Hmm. I don’t know how, not as a general thing — I don’t think I’d even heard of DJVU before — but in this case, inspired by your reply, I found a late version pdf as well as one converted from DJVU (which was less readable).

        However, my impression so far is that Aluffi’s use of Category Theory makes everything much more complicated and confusing than it ought to be. For instance, after he first defines ‘group’, in Joke 1.1, as ‘a groupoid with a single object’, the reader soon encounters this: ‘… and this set carries all the information about G. Call this set G.’ What?

        First you have to notice that the two capital ‘G’s are in different fonts, but even then which set is ‘this set’, to be called italic-serif-G? The Joke even has to be qualified in a footnote that says ‘the group is not the groupoid G, but rather the set of isomorphisms in G, endowed with the operation of composition of morphisms.’ So a group is a set of isomorphisms, so endowed?

        While looking around, I also discovered some aspects of the Category Theory ‘ideology’ I hadn’t known before on a Mathoverflow “categories- first-or- categories-last- in-basic-algebra” page. There seems to be a view that Category Theory is about modelling processes and change. For instance:

        “… the basic thing that one wants to model is a process. … a thing is not defined by what it is (object) but what it does (what category it is in).”

        “About everything in physics deals with processes and change and yet there seems to be very little push to spread the categorical lingua. Relativity screams category theory (equivalent views of the world in different frames yet non identical), the standard model’s soul is categorical (groups, tensor structures of representations, etc…).”

        It never would have occurred to me to think that what category something was in didn’t say what it was but instead what it did. I wonder how that’s meant to fit with way Leinster chapter 3 connects Category Theory to types.

  2. Here is a Sunday morning story about Mac Lane and Birkhoff’s Algebra, a book quite different from the earlier Survey: At the recommendation of a relative, I got a copy and started reading – at the age of 17! Got stuck by about page 120 because the construction of a quotient algebra had no motivation for me. Much later I told this to Garrett Birkhoff in person. He started laughing, something he didn’t do at all otherwise, and said: “That was Saunders’s book!” This just to say that the categorical treatment was not at all his line.

  3. Philip Papayannopoulos

    Dear Peter,

    You could never imagine how many times I have wished you’d also create a Teach Yourself Mathematics guide (along with the fantastic TYL one). I understand of course that this would be an almost impossible (or at least Herculean) task, but this is what wishes are for, right?

    Well, I guess I could take the above post as a step towards this direction?

    1. Thanks for the nice words — though I think I’ll resist your invitation to emulate Hercules (which labour, I wonder … battling the nine-headed Lernaean Hydra I guess!).

      1. Philip Papayannopoulos

        Yes, I think this is the most relevant one (or, perhaps even worse, it could actually be more than one labour..).

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