Customer Review

March 19, 2006
Part 1 (pp. 1-70) deals with some classical algebra (the cubic equation, symmetric polynomials, the fundamental theorem of algebra). It is nice to see the classical roots emphasised, but I think this could have been done in a much more structured and efficient manner. The chapter on the cubic is 20 pages long and involve 28 exercises. One is impatient already---where's the freakin' Galois theory? Actually, there is still another hundred pages until the definition of the Galois group.

Part 2 (pp. 71-188) develops the Galois theory. Modern Galos theory is couched in the language of field extensions (chapters 4-5); more precisely, the Galois group of a field extension is the group of automorphisms that keeps the base field fixed (chapter 6), and the key to the theory is the "Galois correspondence" between the structure of this group and structure of the field extension (chapter 7).

Part 3 (pp. 189-309) deals with applications. The standard applications are here of course (solvability by radicals; straightedge-and-compass constructions; finite fields and their polynomials) but there are also some more novel ones: automorphisms in geometry (finite subgroups of linear fractional transformations); the "casus irreducibilis" (it is not always possible to express real roots by real radicals); Gauss's work on roots of unity (Gauss showed the solvability by radicals of x^p-1=0 by constructing radical expressions for primitive roots of the intermediate field extensions); "origami" (constructions using straightedge, compass and paper folding).

Part 4 (pp. 310-508), "Further Topics", is what sets this book apart from the usual books. In chapter 12 we study some early works on Galois theory. Lagrange's work on solvability by radicals was the obituary for purely classical methods, but as so often before the grave site soil proved fertile and from here Galois sprang forth with his brilliant little paper containing virtually all the ideas we have seen so far. But Galois's insights into solvability by radicals go beyond the insolvability of the quintic, as we see in chapter 14. In fact, his paper culminates with the theorem that if f is of degree p and f=0 is solvable by radicals then, for any two roots a,b of f, Q(a,b) is the splitting field of f. Going further calls for more sophisticated group theory--we study the case when f is of degree p^2, alluded to by Galois, which is about understanding the solvable primitive subgroups of S_(p^2). Chapter 13 treats methods for computing Galois groups. Finally, Cox has saved the best for last: chapter 15 is on Abel's suggestive work on the division of the lemniscate by ruler and compass (n-division is possible when n is a product of a power of 2 by distinct Fermat primes, just as in the case of the circle). Abel had the idea to employ an analog of the sine function, which is given as the inverse of an arc length integral. This function is not only periodic like the sine but doubly periodic in the complex plane, and it has not only addition formulas but formulas for complex multiplication, which we use in our proof of Abel's theorem. Throughout the book one has grown sick and tired of Cox's abusive use of exercises -- arguments are often shortened by statements like "in exercise x you will show so-and-so; therefore ...". Cox has made sure to end on a high note in this respect: after much preparation the proof of Abel's theorem is just over two pages, but it contains no less than eight references to exercises.
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