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Fearless Symmetry by Robert Gross, Avner Ash

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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

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