In this last chapter, we look back and relate everything
to our initial goal of solving systems of Z-equations. We
include a few more explicit examples of systems that can
be studied using reciprocity laws with modular forms.
We take note of how this initial goal has shifted some-
what over the course of our journey, and how it has turned
into a quest for reciprocity laws.
After a digression on why mathematics is worthwhile,
or what motivates people to be interested in mathematics,
we end the book with a brief discussion that attempts to
look past the frontier of current research into the still
murky realm of Galois representations.
Now that we have been over the sometimes rocky, sometimes (we
hope) beguiling road of this book—there being no Royal Road to
mathematics—it is time to look back and see where we have been,
and also to take a glimpse of the view in front of us. Our goal
was to give you some idea of the absolute Galois group of Q and
its representations. We deﬁned and explained basic mathematical
concepts needed before we could even begin our discussion: sets,
groups, various number systems, matrices, and representations.
These concepts are not conﬁned to number theory. They are useful
in many parts of mathematics, science, and technology.