EQUATIONS AND VARIETIES 53
In most of this book, we will deal with equations where all the
constants are integers; for example,
− 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
− x +
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,
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
, the set of integers modulo p,for
various primes p.