222 CHAPTER 19
Here,

W

means as usual the absolute value of W (e.g.,

−4

= 4
and

7

= 7). If you are patient enough, you can untangle what this
theorem and all of the preceding deﬁnitions are describing. You will
ﬁnd out that it is quite close to Theorem 7.2, which we used to prove
quadratic reciprocity. Theorem 19.5 gives us a different approach.
To prove quadratic reciprocity starting from Theorem 19.5, we take
clever choices of W, and use the existence of the eigenelement α that
Theorem 19.3 gives us, together with the knowledge of N given by
Theorem 19.5.
In this section, we will be varying W, so we will put a subscript on
the corresponding α given to us by Theorems 19.3 and 19.5, writing
it as α
W
. That is, α
W
is the simultaneous eigenelement depending
on W described in Theorem 19.4. Because N = 4W, we have the
important fact that if a ≡ b (mod 4W), then α
W
(a) = α
W
(b). We will
use this fact several times in our derivation of quadratic reciprocity.
A Derivation of Quadratic Reciprocity
Suppose ﬁrst that W =−1, so that N = 4. Because α
W
(p) =
−1
p
can be thought of as a morphism deﬁned on (Z/4Z)
×
,wecan
conclude that
−1
p
is determined just by the value of p (mod 4).
Because
−1
3
=−1, and
−1
5
= 1, we can deduce the usual
formula for
−1
p
given on page 79. For example, if p ≡ 3 (mod 4),
−1
p
=
−1
3
=−1. See chapter 7, if you have not already
glanced back to refresh your memory.
Next, take W = 2, and we know that α is a morphism deﬁned on
(Z/8Z)
×
. Computation of
2
p
for p = 3, 5, 7, and 17 gives the usual
formula for
2
p
,whichwegaveonpage79.
Next, suppose that p and q are the odd primes that we want to
compare in quadratic reciprocity. First, assume that p ≡ q (mod 4),