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Fearless Symmetry by Robert Gross, Avner Ash

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We now define and explain the central actor of our drama.
The “absolute Galois group of Q is the set of all permuta-
tions of roots of Z-polynomials that preserve addition and
multiplication. We will call it “the Galois group” for short
and denote it by G, because we believe it is the group
par excellence.
Knowledge about G helps us to solve systems of Z-
equations. Curiously, the deeper our knowledge grows, the
more our interest shifts to G itself, away from particular
equations. The big news is when G is used to solve an old
problem, such as Fermat’s Last Theorem. But the “story
behind the news” is G itself, whose shadowy powers are
still being slowly discovered by number theorists.
G is a group of symmetries and has a rich structure—
much of it still unknown. One of the best ways of prob-
ing that structure is to study the representations of G
into standard objects, namely, permutations groups and
matrix groups. Following this chapter and the next, which
provide two deep examples of G’s activity—namely, the
roots of Z-polynomials and the torsion points of elliptic
curves—we will introduce matrices, matrix groups, and
group representations. Then we will have most of the
ingredients we need to finish our story.

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