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

proof.

1

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

1

For sketches of Wiles’s proof that mention the shoulders on which he stood, see

(Hellegouarch, 2002; Singh, 1997; van der Poorten, 1996).

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