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Fearless Symmetry by Robert Gross, Avner Ash

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EQUATIONS AND VARIETIES 61
Finding Roots of Polynomials
The easiest general class of varieties to look at would be those
defined by a single Z-equation in a single variable, for instance,
x
3
+ x 2 = 0. (6.2)
The study of this type of variety is dominated by the concept of the
Galois group (see chapters 8 and 13).
We now review the following terminology:
DEFINITION: If f (x) is a polynomial, the roots of f (x)are
those numbers c so that f (c) = 0.
The roots of f (x) are the solutions to the polynomial equation
f (x) = 0. Now, there is the particularly simple equation x
n
= a,and
we call a solution to it an nth root of a. Sometimes “root” means a
square root or cube root, and so on, and sometimes it means a root
of a more general polynomial. You should be able to tell which root
is meant from the context.
It is helpful to symbolize the polynomial we are studying by
a single letter, say p. If we want to remember the name of the
variable, we can write p(x). For instance, p(x) might denote the
polynomial x
3
+ x 2. Then equation (6.2) can be written p(x) = 0.
This looks like functional notation, and it is. If x is a variable, p(x)
just stands for the polynomial p,butifa is a number, p(a)stands
for the number you get by substituting a for x in p. For instance, if
p(x) = x
3
+ x 2thenp(0) = 0
3
+ 0 2 =−2, p(1) = 1
3
+ 1 2 = 0,
and in general, p(a) = a
3
+ a 2.
For example, consider the variety of solutions, call it S,tothe
equation p(x) = 0, where p(x) is the polynomial discussed in the pre-
ceding paragraph. That is, if A is any number system, describe the
set S(A), which is the set of all elements a of A such that p(a) = 0.
We can use simple algebra and some guessing to find S(Z): We
have just seen that p(1) = 0. So S(A) contains the number 1. High-
school algebra now tells us that we can divide p(x)byx 1andwe
are guaranteed that it will go in without remainder. Doing that we
get the quotient x
2
+ x + 2.

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