REPRESENTATIONS 11

We could alter the counting process, and stipulate that a mor-

phism be a one-to-one correspondence from the source to a subset of

the target. But that would allow us to count the three oranges on

our desk as “19, 3, 55,” for example, which is useless.

Or is it? If that is our count, then we know that there are three

oranges, because {19, 3, 55} is a set of three numbers. But how do

we know how many numbers are in the set {19, 3, 55}? We still have

to count them, so this technique has not helped us.

Suppose that we require the count to go in order of size. Then

the above example is invalid, but “3, 19, 55” is valid. As always ,

knowing the last number in the count is the point. In this case,

we would then know that the source has at most 55 objects. This

leads to the concept of less than or equal. We could now generate

the science of inequalities by using this kind of morphism.

6

Summary

If A represents B, we have three things: two objects, A and B,

which from now on will be sets, and the relation between them,

which from now on will be a morphism. When A and B have

some additional “structure”—e.g., they are ﬁnite sets, or ordered

sets—and we restrict the possible morphisms from A to B to have

something to do with that structure—e.g., morphisms must be one-

to-one correspondences, or order-preserving functions—then the

existence of a representation from A to B gives us some information

about A in terms of the standard object B—e.g., we can ﬁnd out

how many elements are in A, or at most how many elements are

in A. Another example of adding structure to a set, allowing a more

profound study of that set, is given by the sets of permutations to

which we will add a group structure;seechapter3.

In this book we explore some very explicit examples of represen-

tations. The things we consider are always mathematical objects

such as sets, groups, matrices, and functions between them. We

6

For a nice discussion of counting and its extension t o inﬁnite sets, see One, Two,

Three ...Inﬁnity (Gamow, 1989).

12 CHAPTER 1

show you how this works in detail in one particular case that

we develop throughout the book and that gets us to our goal:

mod p linear representations of Galois groups. We explain how

these representations help to clarify the general problem of solving

systems of polynomial equations with integer coefﬁcients, and how

they can sometimes lead to deﬁnitive results in this area.

Besides the representations discussed in this book, there are

many other kinds of representation theories used today in math-

ematics. Representation theory is often needed to formulate inter-

esting problems, as well as to solve them.

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