158 CHAPTER 14
that the whole universe, and every detail in it, is the expression
of a “One,” a single entity that somehow causes everything else?
Unlike philosophers, mathematicians to some extent get to invent
or arrange their universe the way they like. It is certainly useful
to group similar things together. Sometimes, this causes one large
object to form, which becomes a useful tool for looking at all the
individual things. This is one way to look at G, the absolute Galois
group of Q.
What is the connection between G and the Galois group G(f )
of a given polynomial f (x)? There is a morphism from G to G(f )
called the restriction morphism. It is the crucial glue that holds all
the possible G(f )’s together and welds them into G,or,conversely,
that allows us to go from G, which is an inﬁnite group, to the more
understandable realm of the individual ﬁnite G(f )’s.
The world is nothing more than all the individuals and neigh-
borhoods and countries that comprise it, ﬁtted together with in-
numerable relationships of different kinds. In mathematics, the
relationships are less varied and more mechanical. This difference
makes mathematics easier, or at least more reliable, than politics.
The basic relationship between G and G(f ), the restriction mor-
phism, is a sort of passport that guarantees that G(f ) ﬁts into G.
Here is how to describe the restriction morphism: Remember
is the set of all those complex numbers that are the roots of
Z-polynomials. And G is the group of all one-to-one correspondences
with itself that preserve all the arithmetic operations of
addition, subtraction, multiplication, and division. What is G(f )?
It consists of those permutations of the roots of f (x) that preserve
the arithmetic operations in the ﬁeld of numbers Q(f ) generated by
Now, as we saw in chapter 8, any element γ in G will permute the
roots of any given Z-polynomial. Of course, γ is very busy permut-
ing all the roots of all the Z-polynomials at the same time . But if we
focus in on f (x), we see that, in particular, γ is permuting its roots.
Of course, because γ preserves all the arithmetic operations among
all the algebraic numbers, in particular γ preserves the arithmetic
operations in the ﬁeld of numbers generated by the roots of f (x). In
other words, we can restrict our attention to see how γ acts on the