52 CHAPTER 6

Soon after Descartes, people began to follow his program—with

many changes, of course. One goal was to discover new scientiﬁc

truths. Isaac Newton and Gottfried Leibniz, among others, found

that algebra did not sufﬁce to solve all of their scientiﬁc problems—

they had to invent calculus.

2

Another goal, initiated by Fermat,

shortly after Descartes, was the revival of pure contemplation of

equations among whole numbers just for fun, for curiosity, or for the

greater glory of God. It is this branch of mathematics that contains

the representation theory, the Galois theory, and the number theory

we want to explore in this book.

Even the part of number theory devoted to the investigation of

integers soon turned out to require the use of calculus and other

parts of mathematics. But we will restrict ourselves in this book to

ideas that can be explained using only algebra.

Z-Equations

In German, the word for number is zahl (which is cognate with

our word “tell,” as in “teller who counts money at the bank”).

Therefore, it is customary to use the symbol Z to denote the set

of all integers, namely {0, 1, −1, 2, −2, 3, −3, ...}. We also need to

consider fractions:

DEFINITION: A rational number is any number that can be

expressed as the ratio of two integers. For example, 0, 44, and

−

3

7

are rational numbers, whereas π and

√

2arenot.Real

numbers that are not rational are called irrational numbers.

A real number is rational if and only if it is a terminating decimal

or a repeating decimal. All other decimals are “irrational” real

numbers. The set of all rational numbers is usually denoted by the

symbol Q.

2

You can read the history of physics and how it spurred the development of mathematics

in books such as (Ball, 1960; Boyer, 1991; Smith, 1958; Struik, 1987).

EQUATIONS AND VARIETIES 53

In most of this book, we will deal with equations where all the

constants are integers; for example,

y

2

= x

3

− 3x + 14. (6.1)

The experts call these equations deﬁned over the integers. We will

call them “Z-equations” for short:

DEFINITION: A Z-equation is an equality of polynomials

with integer coefﬁcients.

Note that if you have an equality of polynomials with ratio-

nal number coefﬁcients, you can transform it into an equivalent

Z-equation by multiplying both sides through by a common denomi-

nator. For example, the equation

y

2

3

=

x

3

3

− x +

14

3

can be transformed

into (6.1) by multiplying both sides by 3.

One of the main problems in number theory is ﬁnding and

understanding all solutions of Z-equations. A solution to an equa-

tion can be exempliﬁed using (6.1). We say that x = 2, y = 4isa

solution because if we replace x by 2 and y by 4 on both sides of the

equation and do the arithmetic, we get 16 = 16, which is true. We

can talk about the “solution set S” to a given Z-equation. This is the

set whose elements are all the solutions of that equation.

Wait a minute. Here’s another solution to equation (6.1): x = 1,

y =

√

12. It is a solution, but the y-value is not an integer, and not

even a rational number. Is it in the solution set S?

To quote Humpty Dumpty, “It all depends on which is to be

master.” We can decide what values we will allow the unknowns

to take. If we decide that S will be the set of real-number solutions,

then we will allow x = 1, y =

√

12 into S, but if we want S to contain

only rational-number or integral solutions, we will not.

We need a good notation to clarify in any given context which

solutions are allowed. We do this by symbolizing the sets of possible

kinds of numbers that we are interested in. We let Z be the set of

integers, Q the set of rational numbers, R the set of real numbers,

and C the set of complex numbers. There are other number systems

we will consider, too, especially F

p

, the set of integers modulo p,for

various primes p.

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