QUADRATIC RECIPROCITY II: THE PROOFS
PETE L. CLARK
We shall prove the Quadratic Reciprocity Law and its “second supplement.”
1.
Preliminaries on congruences in cyclotomic rings
For a positive integer
n
, let
ζ
n
=
e
2
πi
n
be a primitive
n
th root of unity, and let
R
n
=
Z
[
ζ
n
] =
{
a
0
+
a
1
ζ
n
+
. . .
+
a
n

1
ζ
n

1
n

a
i
∈
Z
}
.
Recall that an
algebraic integer
is a complex number
α
which satisfies a monic
polynomial relation with
Z
coefficients: there exist
n
and
a
0
, . . . , a
n

1
such that
α
n
+
a
n

1
α
n

1
+
. . .
+
a
1
α
+
a
0
.
We need the following purely algebraic fact:
Proposition 1.
The algebraic integers form a subring of the complex numbers.
This amounts to showing that if
α
and
β
are algebraic integers, then
α
+
β
and
α
·
β
are algebraic integers (which is plausible but not so trivial to prove). We have
relegated the proof to [Integral Elements and Extensions].
Let
p
be a prime number; for
x, y
∈
R
n
, we will write
x
≡
y
(mod
p
) to mean
that there exists a
z
∈
R
n
such that
x

y
=
pz
. Otherwise put, this is congruence
modulo the principal ideal
pR
n
of
R
n
.
Since
Z
⊂
R
n
, if
x
and
y
are ordinary integers, the notation
x
≡
y
(mod
p
)
is ambiguous: interpreting it as a usual congruence in the integers, it means that
there exists an integer
n
such that
x

y
=
pn
; and interpreting it as a congruence
in
R
n
, it means that
x

y
=
pz
for some
z
∈
R
n
. The key technical point is that
these two notions of congruence are in fact the same:
Lemma 2.
If
x, y
∈
Z
and
z
∈
R
n
are such that
x

y
=
pz
, then
z
∈
Z
.
Proof: Just dividing by
p
, we find that the complex number
z
=
x

y
p
is visibly an
element of
Q
. Now we need the fact, proved in Handout 2, that the only algebraic
integers which are rational numbers are the usual integers. (This is the reason why
we needed to assert that every element of
R
n
was an algebraic integer.)
To prove the second supplement we will take
n
= 8.
To prove the QR law we
will take
n
=
p
an odd prime. These choices will be constant throughout each of
the proofs so we will abbreviate
ζ
=
ζ
8
(resp.
ζ
p
) and
R
=
R
8
(resp.
R
p
).
2.
Proof of the Second Supplement
Put
ζ
=
ζ
8
, a primitive eighth root of unity and
R
=
R
8
=
Z
[
ζ
8
]. We have:
0 =
ζ
8

1 = (
ζ
4
+ 1)(
ζ
4

1)
.
1
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2
PETE L. CLARK
Since
ζ
4
= 1 (primitivity), we must have
ζ
4
+ 1 = 0. Multiplying by
ζ

2
we get
ζ
2
+
ζ

2
= 0
.
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 Spring '11
 Staff
 Number Theory, Congruence, Algebraic number theory, quadratic reciprocity law

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