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Fearless Symmetry by Robert Gross, Avner Ash

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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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