In this culminating chapter, we gather the fragmentary
references to Fermat’s Last Theorem from the preceding
chapters and show how the ideas of Galois representa-
tions and reciprocity laws played a central role in its proof.
We also go beyond Fermat’s Last Theorem to consider
similar problems, called “generalized Fermat equations”
and mention that the same ideas can be used to approach
them also. As the equations get more complex, however,
new ideas, outside the scope of this book, will probably be
necessary to solve them.
This chapter is not meant to be a substitute for any
of the popular books that attempt to explain the proof of
Fermat’s Last Theorem. We will focus on the role played
in the proof by the ideas in this book. Nor is there room in
this chapter to give an adequate historical account of the
A word of warning: This chapter is necessarily more
difﬁcult and complex than any other in the book. All the
needed ingredients for understanding it, however, have
been introduced in the previous chapters. A t whatever
level of detail you choose to read it, we hope you will get
For sketches of Wiles’s proof that mention the shoulders on which he stood, see
(Hellegouarch, 2002; Singh, 1997; van der Poorten, 1996).