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Fearless Symmetry by Robert Gross, Avner Ash

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EQUATIONS AND VARIETIES 65
variables. (Of course, for polynomial Z-equations in one variable,
we can bound the size of the roots and then search a finite domain,
so if there is only one variable, then there is a general method for
finding S(Z).)
On a purely number-theoretical level, leaving philosophy and
logic behind, we also have the famous theorem of Abel and Ruffini:
Unlike quadratic polynomials, for which we can use the quadratic
formula, for polynomials f (x) of degree 5 or greater, there is
no formula involving just addition, subtraction, multiplication,
division, and nth roots (n = 2, 3, 4, ...)thatcansolvef (x) = 0in
general.
Deeper Understanding Is Desirable
Finally, even if we had a Sybil, what good would it do us from
a theoretical point of view? We would like to understand why
these solutions exist, if they do, or why they do not, if they do
not, for particular classes of equations. And we would like to
understand the inner structure of the solution sets, and of the
variety. If we know enough about these things, we can then prove
beautiful theorems and understand why they are true. This is what
mathematics is all about. If we had asked Sybil if Fermat’s Last
Theorem were true before Wiles proved it in 1995, and she said
“Yes, would we be satisfied? No. Although we might stop trying to
prove it false, we would continue to try to prove it true, in order to
understand why it is true.
This is the true role of computer-aided proofs, such as the proof of
the Four Color Theorem.
9
They are like Sibyl. It could be that when
we are convinced, by Sybil or by computer, that a given theorem
is true, we might lose interest in the theorem altogether. On the
other hand, mathematicians are always coming up with new proofs
of old theorems, when those theorems remain interesting and the
new proofs enhance our understanding of them.
9
The Four Color Theorem states that it is always possible to color any map with four
colors in such a way that n o two neighboring countries will have the same color. The
proof uses a computer to check that many types of maps can be colored in this way.
66 CHAPTER 6
To go back to a single Z-equation of degree d in a single variable,
defining a variety S:IfA is any field, we can prove that S(A)can
never have more than d solutions in it. (Basically we just keep
dividing by x a for various solutions a.) It is quite interesting
to study S(A) simply as a set. What structure does a finite set
have? Only one thing: How many elements are in it. But knowing
how many elements are in S(A) tells us if there are none, exactly
one, or more than one solution. These can be burning questions.
The security of the credit card number that you just entered on
your computer when you made a purchase over the Internet could
conceivably depend on the answer to such a question.
The amazing discovery of Galois is that there is more structure
to S(A). As we shall see, S(A) is not just a set; it is the basis
for defining a representation of a certain group, called the Galois
group. Even if we only wanted to know how many elements S(A)
has, in many cases this is a difficult question that can only be
approached (as far as anyone today knows) via the Galois group and
its representations. And the same thing is true about Z-varieties
of all kinds. That finally leads us to the central subject of this
book: Galois groups and their representations. But before we go
on to Galois groups, we want to look at some very simple, very
interesting, and very important Z-varieties in the next chapter.
Then, after we explain Galois groups, we will look at another series
of very interesting and very important, though not so very simple,
Z-varieties: elliptic curves. These two kinds of varieties will give us
some of our main examples to help us understand Galois groups
and their representations.

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