QUADRATIC RECIPROCITY REVISITED 217
Let us return to one-dimensional representations, and show how
we can use the concept of a generalized reciprocity law to deduce
quadratic reciprocity. The black box associated with the Galois
representations we consider in this chapter is constructed from
an integer N > 1 and a set of functions F
with some special
properties. First we have to make the following deﬁnition:
DEFINITION: Let N be an integer greater than 1. Then
is the set of all integers b modulo N that have
multiplicative inverses modulo N.
For example, 3 is in (Z/10Z)
because 3 · 7 ≡ 1 (mod 10), but 5
is not in (Z/10Z)
because there is no solution to the congruence
equation 5x ≡ 1 (mod 10). If b is in (Z/NZ)
, we write b
element in (Z/NZ)
that satisﬁes the congruence bb
≡ 1 (mod N).
A theorem that is proved in most introductory number the-
ory courses says that the elements of (Z/NZ)
are the numbers
between 1 and N − 1 that have no factor in common with N other
Now we can deﬁne the set F
DEFINITION: Let N be an integer greater than 1. Then F
stands for the set of all functions from (Z/NZ)
If α is a function in F
, and if b is an element of (Z/NZ)
as usual with functional notation, we write α(b) for the complex
number in C
that the rule α assigns to b.
The set F
contains the black box. To describe these black boxes,
we need to consider certain functions inside of this set.
DEFINITION: Let N be an integer greater than 1. A
simultaneous eigenelement in F
is a function α in F
Recall that C
denotes the set of all nonzero complex numbers.