Advanced Modern Algebra

Algebra is used by virtually all mathematicians, be they analysts, combinatorists, computer scientists, geometers, logicians, number theorists, or topologists. Nowadays, everyone agrees that some knowledge of linear algebra, groups, and commutative rings is necessary, and these topics are introduced in undergraduate courses. We continue their study.

This book can be used as a text for the first year of graduate algebra, but it is much more than that. It can also serve more advanced graduate students wishing to learn topics on their own; while not reaching the frontiers, the book does provide a sense of the successes and methods arising in an area. Finally, this is a reference containing many of the standard theorems and definitions that users of algebra need to know. Thus, the book is not only an appetizer, but a hearty meal as well.

Let me now address readers and instructors who use the book as a text for a beginning graduate course. If I could assume that everyone had already read my book, A First Course in Abstract Algebra, then the prerequisites for this book would be plain. But this is not a realistic assumption; different undergraduate courses introducing abstract algebra abound, as do texts for these courses. For many, linear algebra concentrates on matrices and vector spaces over the real numbers, with an emphasis on computing solutions of linear systems of equations; other courses may treat vector spaces over arbitrary fields, as well as Jordan and rational canonical forms. Some courses discuss the Sylow theorems; some do not; some courses classify finite fields; some do not.

To accommodate readers having different backgrounds, the first three chapters contain many familiar results, with many proofs merely sketched. The first chapter contains the fundamental theorem of arithmetic, congruences, De Moivre’s theorem, roots of unity, cyclotomic polynomials, and some standard notions of set theory, such as equivalence relations and verification of the group axioms for symmetric groups. The next two chapters contain both familiar and unfamiliar material. “New” results, that is, results rarely taught in a first course, have complete proofs, while proofs of “old” results are usually sketched. In more detail, Chapter 2 is an introduction to group theory, reviewing permutations, Lagrange’s theorem, quotient groups, the isomorphism theorems, and groups acting on sets. Chapter 3 is an introduction to commutative rings, reviewing domains, fraction fields, polynomial rings in one variable, quotient rings, isomorphism theorems, irreducible polynomials, finite fields, and some linear algebra over arbitrary fields. Readers may use “older” portions of these chapters to refresh their memory of this material (and also to see my notational choices); on the other hand, these chapters can also serve as a guide for learning what may have been omitted from an earlier course (complete proofs can be found in A First Course in Abstract Algebra). This format gives more freedom to an instructor, for there is a variety of choices for the starting point of a course of lectures, depending on what best fits the backgrounds of the students in a class. I expect that most instructors would begin a course somewhere in the middle of Chapter 2 and, afterwards, would continue from some point in the middle of Chapter 3. Finally, this format is convenient for the author, because it allows me to refer back to these earlier results in the midst of a discussion or a proof. Proofs in subsequent chapters are complete and are not sketched.

I have tried to write clear and complete proofs, omitting only those parts that are truly routine; thus, it is not necessary for an instructor to expound every detail in lectures, for students should be able to read the text.

When I was a student, Birkhoff and Mac Lane’s A Survey of Modern Algebra was the text for my first algebra course, and van der Waerden’s Modern Algebra was the text for my second course. Both are excellent books (I have called this book Advanced Modern Algebra in homage to them), but times have changed since their first appearance: Birkhoff and Mac Lane’s book first appeared in 1941, and van der Waerden’s book first appeared in 1930. There are today major directions that either did not exist over 60 years ago, or that were not then recognized to be so important. These new directions involve algebraic geometry, computers; homology, and representations (A Survey of Modern Algebra has been rewritten as Mac Lane-Birkhoff, Algebra, Macmillan, New York, 1967, and this version introduces categorical methods; category theory emerged from algebraic topology, but was then used by Grothendieck to revolutionize algebraic geometry).

Here is a more detailed account of the later chapters of this book.

Chapter 4 discusses fields, beginning with an introduction to Galois theory, the interrelationship between rings and groups. We prove the insolvability of the general polynomial of degree 5, the fundamental theorem of Galois theory, and applications, such as a proof of the fundamental theorem of algebra, and Galois’s theorem that a polynomial over a field of characteristic 0 is solvable by radicals if and only if its Galois group is a solvable group.

Chapter 5 covers finite abelian groups (basis theorem and fundamental theorem), the Sylow theorems, Jordan-Hölder theorem, solvable groups, simplicity of the linear groups PSL(2, k), free groups, presentations, and the Neilsen-Schreier theorem (subgroups of free groups are free).

Chapter 6 introduces prime and maximal ideals in commutative rings; Gauss’s theorem that R x is a UFD when R is a UFD; Hilbert’s basis theorem, applications of Zorn’s lemma to commutative algebra (a proof of the equivalence of Zorn’s lemma and the axiom of choice is in the appendix), inseparability, transcendence bases, Lüroth’s theorem, affine varieties, including a proof of the Nullstellensatz for uncountable algebraically closed fields (the full Nullstellensatz, for varieties over arbitrary algebraically closed fields, is proved in Chapter 11); primary decomposition; Gröbner bases. Chapters 5 and 6 overlap two chapters of A First Course in Abstract Algebra, but these chapters are not covered in most undergraduate courses.

Chapter 7 introduces modules over commutative rings (essentially proving that all R-modules and R-maps form an abelian category); categories and functors, including products and coproducts, pullbacks and pushouts, Grothendieck groups, inverse and direct limits, natural transformations; adjoint functors; free modules, projectives, and injectives.

Chapter 8 introduces noncommutative rings, proving Wedderburn’s theorem that finite division rings are commutative, as well as the Wedderburn-Artin theorem classifying semisimple rings. Modules over noncommutative rings are discussed, along with tensor products, flat modules, and bilinear forms. We also introduce character theory, using it to prove Burnside’s theorem that finite groups of order p^mqⁿ are solvable. We then introduce multiply transitive groups and Frobenius groups, and we prove that Frobenius kernels are normal subgroups of Frobenius groups.

Chapter 9 considers finitely generated modules over PIDs (generalizing earlier theorems about finite abelian groups), and then goes on to apply these results to rational, Jordan, and Smith canonical forms for matrices over a field (the Smith normal form enables one to compute elementary divisors of a matrix). We also classify projective, injective, and flat modules over PIDs. A discussion of graded k-algebras, for k a commutative ring, leads to tensor algebras, central simple algebras and the Brauer group, exterior algebra (including Grassman algebras and the binomial theorem), determinants, differential forms, and an introduction to Lie algebra.

Chapter 10 introduces homological methods,beginning with semidirect products and the extension problem for groups. We then present Schreier’s solution of the extension problem using factor sets, culminating in the Schur-Zassenhaus lemma. This is followed by axioms characterizing Tor and Ext (existence of these functors is proved with derived functors), some cohomology of groups, a bit of crossed product algebras, and an introduction to spectral sequences.

Chapter 11 returns to commutative rings, discussing localization, integral extensions, the general Nullstellensatz (using Jacobson rings), Dedekind rings, homological dimensions, the theorem of Serre characterizing regular local rings as those noetherian local rings of finite global dimension, the theorem of Auslander and Buchsbaum that regular local rings are UFDs.

Each generation should survey algebra to make it serve the present time.

It is a pleasure to thank the following mathematicians whose suggestions have greatly improved my original manuscript: Michael Barr, Daniel Bump, Heng Huat Chan, Ulrich Daepp, Boris A. Datskovsky, Keith Dennis, Vlastimil Dlab, Sankar Dutta, David Eisenbud, E. Graham Evans, Jr., Daniel Flatly Jeremy J. Gray, Daniel Grayson, Philllip Griffith, William Haboush, Robin Hartshorne, Craig Huneke, Gerald J. Janusz, Carl Jockusch, David Leep, Marcin Mazur, Leon McCulloh, Emma Previato, Eric Sommers, Stephen V. Ullom, Paul Vojta, William C. Waterhouse, and Richard Weiss.

Joseph Rotman