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Fearless Symmetry by Robert Gross, Avner Ash

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QUADRATIC RECIPROCITY REVISITED 217
Simultaneous Eigenelements
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
N
with some special
properties. First we have to make the following definition:
DEFINITION: Let N be an integer greater than 1. Then
(Z/NZ)
×
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
1
for the
element in (Z/NZ)
×
that satisfies the congruence bb
1
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
than 1.
Now we can define the set F
N
:
DEFINITION: Let N be an integer greater than 1. Then F
N
stands for the set of all functions from (Z/NZ)
×
C
×
.
1
If α is a function in F
N
, and if b is an element of (Z/NZ)
×
, then
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
N
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
N
is a function α in F
N
with
1
Recall that C
×
denotes the set of all nonzero complex numbers.

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