Algebra 1

I think this set of books (with a third volume evidently in the chute) is a very good idea, pedagogically speaking. Lal intends to provide a complete course, or sequence of courses, from the early undergraduate to the (post)graduate level, the subject being pretty much all things algebra. Accordingly it all begins with logic and set theory, very quickly followed by the nuts and bolts of groups, rings and fields, and even some number theory. Indeed, Volume I closes with “Arithmetic in Rings,” in which Lal takes the reader from division in rings through PIDs and Euclidean domains to the Chinese Remainder Theorem; he finishes with UFDs. So, Volume I fits well into the ordinary scheme of things in an undergraduate course, insofar as it leads the student to the point of being able to start a course in algebraic number theory, presumably as an advanced undergraduate.

Well, we need something of a caveat, of course, namely, the warning that linear algebra is missing. But that’s precisely where Volume II comes in: Lal goes from linear algebra as done just about everywhere (but he does it well …), through Galois theory, through fundamental representation theory , to the important but occasionally neglected subject of group extensions and the Schur multiplier (so, certainly Lal gets kudos here). Volume II is very heavy on linear algebra, including both the computational aspects (lots of stuff on massaging matrices) and the critical theory of linear spaces and their morphisms. It’s all there, including basis change, rank-nullity, inner products, eigenvalues and eigenvectors, the theory of modules over a PID (“Canonical Forms, Jordan and Rational Forms”), and so on. Lal even deals with “General Linear Algebra,” providing something of a prelude to homological algebra; we get, e.g., free, projective, and injective modules, tensor products, exterior algebras: a lot of good stuff (which I first saw in Cartan-Eilenberg, and therefore always think of as belonging to homological algebra).

The biggish eighth chapter of Volume II is devoted to Galois theory, and it’s appropriate to observe that in the best case scenario, the advanced undergraduate mentioned above, who, having experienced the epiphany that number theory is the way to go, wants to sign up for the aforementioned algebraic number theory course, should see a decent amount of Galois theory first. The coverage of this material is very good: it is very accessible, covers the needed material, and ends with “boots on the ground,” i.e. the business of solving particular polynomials via Galois theory. Lal closes the chapter with a discussion of the cubic and Cardano’s solution, followed by a brief treatment of Ferrari’s method for going at the general quartic: stuff everyone should know, or at least know about. Very cool.

The books are well-written (modulo a few typos I saw that are pretty much harmless) and complete, given their objectives (which they meet well). Lal provides good exercise sets that cover the appropriate relative difficulties spectrum. For example, getting back to his discussion at the end of Chapter 8, here are the problems he appends:

8.8.1    Solve … \(X^3-3X^2+2X+1=0\).

8.8.2    Solve … \(X^4+4X-1=0\).

8.8.3    Find a polynomial of degree \(7\) over \(\mathbb{Q}\) whose Galois group is \(S_7\)

A nice selection, I think.

Lal’s Volume III is evidently geared to graduate work; he mentions that it will cover such things as commutative algebra, Lie algebras, more advanced representation theory, and even Chevalley groups. Fair enough. But I should like to note that I’d recommend more algebraic number theory to the corresponding graduate students. I guess I’m apt to recommend that as a sort-of default move, however, while in reality not everyone experiences the epiphany I mentioned above. C’est la vie. 

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.