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Fearless Symmetry by Robert Gross, Avner Ash

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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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