There are two more major ingredients we need before
we can pull everything together and explain how linear
representations of Galois groups allow us to generalize
quadratic reciprocity into a vast theory of generalized
reciprocity laws (much of which is still only conjectural).
Those two things are characters and Frobenius elements.
Characters are functions attached to linear representa-
tions of groups. That is one reason to use linear represen-
tations instead of permutation representations. In fact,
characters are only one piece of the whole characteristic
polynomial of a linear representation. Characters are
sufﬁcient for us in this book, although for a full exposition
of generalized reciprocity laws we would need the whole
Characters have no exclusive relationship to number
theory. They can be deﬁned for any matrix representation
of any group. On the other hand, Frobenius elements
belong to Galois groups and thus live only in a number-
We will explain characters in this chapter and Fro-
benius elements in the next chapter.