Algebra with Galois Theory

As a mathematician, and more concretely as an algebraist and a professor, I was very excited to browse Emil Artin's "Algebra with Galois theory". What a great opportunity to take a peek at the teaching and exposition style of a great mathematician!

This thin book (126 pages) contains the lecture notes for a course in algebra, taught by Emil Artin and recorded in writing by Albert A. Blank. These notes were first published in 1947 and became very popular as a textbook among instructors and students alike. Due to its popularity, the Courant Institute decided to reprint the notes in 2007.

These lecture notes first appeared in print under the title Modern Higher Algebra: Galois Theory, but, as the editors remark in a note: "because what was in 1947 'modern' has now become standard, and what was then 'higher' has now become foundational, we have retitled this volume Algebra with Galois theory."

Indeed, a reader today would find that the content is neither "modern" nor "higher," but classical and standard. In fact, the contents of the volume are very light in comparison to the typical syllabus for an algebra course today. However, it is clear that these notes are not meant as a thorough nitty-gritty introduction to algebra. Rather, they are meant to be an overview of algebra that motivates the definitions of groups, rings, fields, etc, as the right tools to solve some very interesting problems, such as: (a) what ruler and compass constructions are possible, or (b) the unsolvability of the quintic by radicals.

With solving (a) and (b) as his goals, Artin only covers the very basic tools that are needed to prove these classical theorems. The text begins with a brief chapter on group theory which simply covers the definitions of group, subgroup and the proof of Lagrange's theorem (the order of a subgroup divides the order of the group). The next chapter defines the concepts of ring (not necessarily commutative), "field" (a ring with multiplicative inverses) and "commutative field" (a field where multiplication is commutative), followed by a brief introduction to linear algebra: vector spaces, dimension and systems of linear equations.

Chapter 3 defines polynomial rings and the basic operations on polynomials (he covers irreducibility criteria and cyclotomic polynomials in Chapter 6). Then moves on to define ideals of rings in general, discusses gcds and factorizations into primes, and shows that a principal ideal domain is a unique factorization domain.

Chapters 4 and 5 constitute the central core of the lecture notes. Since Artin constructs extensions of a field F using quotients of the polynomial ring F[x] by a maximal ideal, he begins by studying congruences in rings. And then defines extension fields, isomorphisms of fields, finite fields, splitting fields, the automorphism group of an splitting field, the concept of characteristic, the concept of separability for polynomials, the degree of an extension, group characters with values on a field, "automorphic" groups (groups generated by a finite number of automorphisms of a field) and the fixed fields by these groups, and, finally, the fundamental theorem of Galois theory.

All the background material has now been covered and, in the final chapter, Artin utilizes Galois theory to show what polygons can and cannot be constructed with a ruler and compass, and to show what equations can be solved by radicals. In particular, Abel's theorem is proved: a general polynomial equation of degree n > 4 cannot be solved by radicals.

These notes are very well-written, reader friendly and student friendly — they contain lots of detailed examples and also a very healthy number of exercises. Even though the approach is interesting, and the text did satisfy my curiosity to peek into Artin's style, I have my doubts about the usefulness of these notes nowadays — other than sentimental reasons. Today, most algebra courses cover each topic in much greater depth (there are only 9 pages on group theory!) and they are expanded into two or even three different semesters. In fact, I don't know of any mathematics department that would dare to cover groups, rings, fields and Galois theory in one single course! However, if a department wanted to create an alternative course, for those who do not wish to take a year-long algebra sequence, the material in this book could constitute a very nice syllabus.

Álvaro Lozano-Robledo is H. C. Wang Assistant Professor of mathematics at Cornell University.