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 irr ...
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
In algebra, a quartic function is a function of the form
:f(x)=ax^4+bx^3+cx^2+dx+e,
where ''a'' is nonzero,
which is defined by a polynomial of degree four, called a quartic polynomial.
A ''quartic equation'', or equation of the fourth degre ...
and
octic reciprocity laws. 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
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Algebraic number theory
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