In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, 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 reciprocity law, quadratic reciprocity, determines when an irr ...
relating the residues of 8th powers
modulo primes
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, analogous to the
law of quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...
,
cubic reciprocity
Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence ''x''3 ≡ ''p'' (mod ''q'') is solvable; the word "reciprocity" comes from the form of ...
, and
quartic reciprocity
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence ''x''4 ≡ ''p'' (mod ''q'') is solvable; the word "reciprocity" comes from the form o ...
.
There is a
rational reciprocity law for 8th powers, due to Williams. Define the symbol
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
Let ''p'' = ''a''
2 + ''b''
2 = ''c''
2 + 2''d''
2 and ''q'' = ''A''
2 + ''B''
2 = ''C''
2 + 2''D''
2, with ''aA'' odd. Then
:
See also
*
Artin reciprocity
Artin may refer to:
* Artin (name), a surname and given name, including a list of people with the name
** Artin, a variant of Harutyun
Harutyun ( hy, Õ€Õ¡Ö€Õ¸Ö‚Õ©ÕµÕ¸Ö‚Õ¶ and in Western Armenian Õ…Õ¡Ö€Õ¸Ö‚Õ©Õ«Ö‚Õ¶) also spelled Haroutioun, Harut ...
*
Eisenstein reciprocity In algebraic number theory Eisenstein's reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubic reciprocity law to residues of higher powers. It is one of the earliest and simplest of the higher reciprocity ...
References
*
*
Theorems in algebraic number theory
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