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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