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Fearless Symmetry by Robert Gross, Avner Ash

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FERMAT’S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS 243
some idea of how the proof of Fermat’s Last Theorem uses
the “fearless symmetries” of the Galois group that have
been our theme.
The Three Pieces of the Proof
In the last chapter, we gave our last look at reciprocity laws per se.
Now we will show how these laws can be applied to prove Fermat’s
Last Theorem (referred to as FLT throughout this chapter) and
similar theorems.
Before Andrew Wiles burst upon the scene, there were a number
of proposed strategies for proving FLT. Let us begin by explaining
the strategy that Wiles ultimately made to work. This strategy is
composed of several difficult theorems plus one big conjecture. The
proof was completed by Wiles and Taylor–Wiles by proving enough
of the big conjecture to make the whole strategy work.
2
Before we start, we review the statement of FLT:
If n is an integer greater than 2, there are no solutions to
the equation x
n
+ y
n
= z
n
, where we require x, y, and z to be
nonzero integers.
The first step is to note that if we can prove this for the exponents
4 and odd prime numbers, then it is true for any n > 2. Why? Given
any integer n > 2, there are two possibilities:
1. If n is divisible by 4, we can write n = 4m, and then the
equation a
n
+ b
n
= c
n
becomes (a
m
)
4
+ (b
m
)
4
= (c
m
)
4
.Ifwe
have shown that there are no solutions with the exponent
4, then this equation has no solutions.
2. If n is not divisible by 4, then it must be divisible by an odd
prime p. Write n = pm, and then the equation a
n
+ b
n
= c
n
becomes (a
m
)
p
+ (b
m
)
p
= (c
m
)
p
. If we have shown that there
are no solutions with the exponent p, then this equation
has no solutions.
2
Wiles collaborated with Richard Taylor to prove one key step in the whole proof. Thus,
the proof of FLT appeared in two papers, one by Wiles (Wiles, 1995) and a shorter one
that was a joint work of Taylor and Wiles (Taylor and Wiles, 1995).

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