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Reciprocity law relating the residues of 8th powers modulo primes
In
number theory , octic reciprocity is a
reciprocity law relating the residues of 8th powers
modulo
primes , analogous to the
law of quadratic reciprocity ,
cubic reciprocity , and
quartic reciprocity .
There is a
rational reciprocity law for 8th powers, due to Williams. Define the symbol
(
x
p
)
k
{\displaystyle \left({\frac {x}{p}}\right)_{k}}
to be +1 if x is a k -th power modulo the prime p and -1 otherwise. Let p and q be distinct primes congruent to 1 modulo 8, such that
(
p
q
)
4
=
(
q
p
)
4
=
+
1.
{\displaystyle \left({\frac {p}{q}}\right)_{4}=\left({\frac {q}{p}}\right)_{4}=+1.}
Let p = a 2 + b 2 = c 2 + 2d 2 and q = A 2 + B 2 = C 2 + 2D 2 , with aA odd. Then
(
p
q
)
8
(
q
p
)
8
=
(
a
B
−
b
A
q
)
4
(
c
D
−
d
C
q
)
2
.
{\displaystyle \left({\frac {p}{q}}\right)_{8}\left({\frac {q}{p}}\right)_{8}=\left({\frac {aB-bA}{q}}\right)_{4}\left({\frac {cD-dC}{q}}\right)_{2}\ .}
See also
References
Lemmermeyer, Franz (2000),
Reciprocity laws. From Euler to Eisenstein , Springer Monographs in Mathematics, Springer-Verlag, Berlin, pp. 289–316,
ISBN
3-540-66957-4 ,
MR
1761696 ,
Zbl
0949.11002
Williams, Kenneth S. (1976),
"A rational octic reciprocity law" ,
Pacific Journal of Mathematics , 63 (2): 563–570,
doi :
10.2140/pjm.1976.63.563 ,
ISSN
0030-8730 ,
MR
0414467 ,
Zbl
0311.10004