- Source: Octic reciprocity
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 = a2 + b2 = c2 + 2d2 and q = A2 + B2 = C2 + 2D2, 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
Artin reciprocity
Eisenstein reciprocity
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