Gauss’s Quadratic Reciprocity Theorem
: NumThy
Jonathan L.F. King
University of Florida, Gainesville FL 326112082, USA
[email protected]
Webpage
http://www.math.ufl.edu/
∼
squash/
30 November, 2011
(at
01:30
)
1
:
Nomenclature.
For odd
D
, use
H
D
to mean
D

1
2
.
(
The
H
is to suggest “Half”.
)
In the sequel,
p
is an odd prime and
S
⊥
p
is
the
“
stridelength
”
;
we
will
walk
around
the
circumference=
p
circle using strides of length
S
.
Use
H
:=
H
p
and
hh
x
ii
:=
hh
x
ii
p
for the
symmetric
residue of integer
x
modulo
p
; so
hh
x
ii
is in
[
H .. H
]
.
Let
≡
mean
≡
p
.
Let
G
=
G
p
(
S
) be the set
of indices
‘
∈
[
1
.. H
]
such
that
hh
‘
·
S
ii
p
is neGative. Letting
P
be the indices
with
hh
‘
·
S
ii
Positive, we have that (
disjointly
)
G
t
P
=
[
1
.. H
]
.
Finally, use
N
=
N
p
(
S
) for the number of “negative”
indices;
N
:=
#
G
.
2
:
Prop’n.
Fix an
S
⊥
p
, with notation from
(1)
.
Then the mapping
(
absolutevalue of symmresidue
)
‘
7→
hh
‘
·
S
ii
,
is a
permutation
of
[
1
.. H
]
. We say that the map
ping
‘
7→ hh
‘
·
S
ii
is a
“
permutation up to sign
”
of
[
1
.. H
]
.
♦
Proof.
Given indices 1
6
‘
6
k
6
H
, we want that
either equality
∓hh
‘
·
S
ii
=
hh
k
·
S
ii
forces
‘
=
k
.
For either choice of sign in
∓
, note that
∓hh
‘
·
S
ii
=
hh
k
·
S
ii
IFF
0
≡
[
k
±
‘
]
·
S
IFF
0
≡
k
±
‘ ,
since
S
⊥
p
. Thus
0
6
k
±
‘
6
2
H <
p
.
Together with
k
±
‘
≡
0, this forces
k
±
‘
to actually
be
zero. Thus the “
±
” is a minus sign, and
k
=
‘
.
3
:
Gauss Lemma.
Fix an odd prime
p
and integer
S
⊥
p
. Then the Legendre symbol
(
S
p
)
satisfies
(
S
p
)
= [
1]
N
.
♦
Proof of Gauss Lemma.
Necessarily
H
Y
‘
=1
hh
‘
·
S
ii ≡
H
Y
‘
=1
‘
·
S
=
H
!
·
S
H
≡
H
!
·
S
p
!
,
4
:
with the last step following from LSThm.
Observe
that
hh
‘
·
S
ii
equals
± hh
‘S
ii
as
‘
isnot/is in
G
.
Prop’n 2, consequently, tells us that LhS(4) can be
written as
H
! times [
1]
N
. Thus RhS(4) equals
H
!
·
(
S
p
)
≡
H
!
·
[
1]
N
.
The
H
!, being coprime to
p
, cancels mod
p
to hand
us congruence
(
S
p
)
≡
[
1]
N
.
An important application is the following.
5
:
TwoisQR Lemma.
Consider an oddprime
p
. Then
2
is a
p
QR IFF
p
≡
8
±
1
.
♦
Abbrev.
An odd integer
D
is
8Near
if
D
≡
8
±
1; it
is
8Far
if
D
≡
8
±
3. (
The names come from being, mod 8,
near/far from zero.
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 Fall '07
 JURY
 Math, Calculus, Number Theory, gauss lemma, Prof. JLF King

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