To be completely honest, I didn't much care for this book when I got about halfway through it. The author was leading me down a path (or was he?), but I couldn't tell where the heck we were going. In the end, he turned on the light switch just in time....and I enjoyed the experience thoroughly.
The book starts with a description of the problem, moves into a history of the mathematics needed to solve the problem (some of it quite technical and confusing, even for this former physicist), and then ends quite nicely with Andrew Wiles and the politics of big-time mathematics. I bet you didn't know big-time mathematics had politics associated with it, did you? For that matter, you probably didn't even know there was such a thing as big-time mathematics.
In short, if you can stick it out through the math part of the book (the middle 100 pages, 50-75% of which is interesting to the casual reader), the author lays out a nice history of mathematics and the task of solving Fermat's Last Theorem.