In number theory, a rational reciprocity law is a
reciprocity law
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f(x) with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an ir ...
involving residue symbols that are related by a factor of +1 or –1 rather than a general root of unity.
As an example, there are rational
biquadratic and
octic reciprocity law
In number theory, octic reciprocity is a reciprocity law
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f(x) with integer coefficients. Recall that first reci ...
s. Define the symbol (''x'', ''p'')
''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 4, such that (''p'', ''q'')
2 = (''q'', ''p'')
2 = +1. Let ''p'' = ''a''
2 + ''b''
2 and ''q'' = ''A''
2 + ''B''
2 with ''aA'' odd. Then
:
If in addition ''p'' and ''q'' are congruent to 1 modulo 8, let ''p'' = ''c''
2 + 2''d''
2 and ''q'' = ''C''
2 + 2''D''
2. Then
:
References
*
*
*
*
Algebraic number theory
{{numtheory-stub