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

characteristic polynomial.

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-

theoretic world.

We will explain characters in this chapter and Fro-

benius elements in the next chapter.

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