Fermat's Last Theorem:

There are no solutions to the following problem with (X, Y, Z) integers

X^p + Y^p + Z^p = 0

$$\neq$$ XYZ 0

$$\geq$$ and a prime

The approach to the proof of Fermat's Last Theorem that is followed by A. Wiles in his recent attempt can be thought of as a particular case of the following tactic.

Suppose (X, Y, Z) is a counter-example to Fermat's Last Theorem.

1. To such a counter-example we attach a representation
   $$\rho_{(X,Y,Z)}^{}$$ : Gal($$\overline{{\Bbb Q}}$$/$$\Bbb$$Q)$$\to$$GL_n($$\Bbb$$F_p)
   Moreover, we have good ramification properties for this representation. For example,
   1. the representation is unramified outside p,
   2. the representation has ``good'' ramification properties at p.
2. The next step is to use our knowledge of Algebraic Number Theory to prove that such representations are impossible.

The proof of Kummer for the case of regular primes can also be reviewed in this light. First of all, Kummer's proof associates to every counter-example (X, Y, Z), a representation

$$\rho$$ : Gal($$\overline{K}$$/K)$$\to$$$$\Bbb$$F_p

where K is cyclotomic field of p-th roots of unity. Next, he gives a way of finding out which primes p are such that we have such a representation. As he showed, there are indeed such primes and thus his proof works only for ``regular'' primes. In section 1 we recall some computations in the cyclotomic field of p-th roots of unity. In section 2 we show how a counter-example to Fermat's Last Theorem (if it exists) can be used to construct a cyclic extension of order p of the cyclotomic field which is unramified everywhere. We review the Class number formula in section 3. Finally, in Section 4 we use this formula to check when such unramified extensions do indeed exist. Most of the material in this note can be found in more detail (though in a more classical presentation) in the book of H. M. Edwards [1]. This re-examination of Kummer's proof was inspired by some remarks made by V. Kumar Murty during his lecture on the work of Wiles at the TIFR. I would like to thank A. Raghuram for his careful reading of the manuscript and numerous suggestions.

We fix a prime p $\geq$ 5 throughout the discussion.