THE GREEKS HAD A NAME FOR IT 175

Prelude to the Next Chapter

Group representations and their characters were ﬁrst studied over

100 years ago. It was seen immediately that characters were very

important for understanding groups and their representations. But

in number theory, the character takes on a new life in the world

of reciprocity laws. If the representation is of a Galois group G(f )

of a ﬁeld Q(f ), then the character can tell us a lot about the

ﬁeld Q(f ), and about how the polynomial f (x) behaves modulo

various primes p. This is especially true if the representation is a

faithful one.

To repeat: If r is a linear representation of a ﬁnite Galois

group G(f ), then the character of r reveals many important and

fascinating number-theoretic properties of f and G(f ). How? We

have to apply it to special elements of G(f ), called Frobenius

elements. We will deﬁne them in the next chapter.

Before going on to the next chapter, though, remember the

restriction morphism of the absolute Galois group G.Ifwehave

a representation r of a ﬁnite Galois group H = G(f ), we can always

compose it (as a function) with the restriction morphism r

H

: G →

H to get a representation of G which has basically the same

information as r. We like to do this, because then we can refer

everything to the same big group G. It will be easiest for us to

discuss the Frobenius elements ﬁrst as elements in G and then

restrict them (via r

H

)toH, and then we can apply r.

Digression: A Fact about Rotations of the Sphere

In a footnote on page 15, we mentioned something that may have

been on your mind ever since. We had an example of two elements

g and h in SO(3) and we showed that g ◦ h = h ◦ g. And then we

mentioned that in fact g ◦ h and h ◦ g will be rotations through the

same number of degrees, just with different axes of rotation.

This can be proved by using traces. For simplicity of notation,

we think of each rotation in SO(3) as being equal to the matrix

in GL(3, R) that represents it, as described on page 145. The key

176 CHAPTER 15

is that g ◦ h and h ◦ g are in the same conjugacy class, because

g ◦ h = h

−1

◦

h ◦ g

◦ h. Therefore, the trace of g ◦ h is the same as

the trace of h ◦ g. Now, a bit of trigonometry will show you that

if A is a rotation of θ

◦

in SO(3), then the trace of A is 1 + 2 cos θ

◦

.

Therefore, if two different elements in SO(3) have the same trace,

then they must be rotations by the same amount.

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