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
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.