We review matrices from the ground up, because they will

play a major role in our story from now on. Our main focus

is matrix multiplication, which will be used as a group

operation in the next chapter, thereby giving us a large

supply of groups of matrices which we understand quite

well. We will then use matrix groups as standard objects—

targets—in Galois representation theory. (This could be a

good time to review chapter 1.)

Matrices and Matrix Representations

We need matrices in order to get to the center of our subject—

representations o f Galois groups. There are two types of represen-

tations we must consider: matrix representations and permutation

representations. We will discuss matrix representations in chap-

ter 12 and permutation representations in chapter 14.

The two kinds of representations are closely related. Because we

deﬁned the Galois group as a group of permutations, one may think

that it would be enough to study permutation representations. But

even if we knew everything about the permutation representations,

we would need the matrix kind in order to formulate some of the

deeper properties of Galois groups and Z-varieties. W e shall see how

this surprising fact plays out in the ﬁnal chapters of this book.

Matrices are important, even if we have no interest in number

theory, let alone Galois groups. They belong to algebra, but they are

Start Free Trial

No credit card required