Strange book. It is somewhere between a popular science book and a textbook. Maybe the best use is for the not number-theorist mathematicians to understand what is going on there.
It is an easy read, at least for the first 2 parts. I understood the new concept quickly and the structures that I did know beforehand are also explained very nicely and quickly. Sometimes their strategy is that say something firm, easy to understand/remember then to admit that it is a lie, the situation is a bit more complicated. It seems to be a useful technique.
So what is achieved in the first 2 parts? Group theory with permutation and matrix representations, elliptic functions, Galois groups, characters. Then it all comes together in reciprocity laws, using Galois representations for understanding the set of solutions of equations… but at this point, in Part 3 it becomes quite steep. New concepts are coming in a pace that full understanding is not possible any more. Understanding Wiles’ proof after reading a thin book is not feasible under normal circumstances. Most probably that is not the aim here, it is rather wetting the appetite for learning more, and that is done well.
An index of symbols would be very useful. The authors keep referring back to previous definitions, reflect on notations, so the omission of this is quite surprising.
The text contains some philosophical reflection on math at some random points, and these are quite good, though I was quite surprised to hear that some people think that mathematics is finished. I never met those people.
The preface for the paperback edition is great fun. It is a pity that that sense of humour is not pursued in the main text.