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 (1777� ...

, 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 In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is t ...

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 In number theory, a rational reciprocity law is a reciprocity law 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 f ...

for 8th powers, due to Williams. Define the symbol

\left(\frac xp\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

\left(\frac pq\right)_4 = \left(\frac qp\right)_4 = +1 .

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 :

\left(\frac pq\right)_8 \left(\frac qp\right)_8 = \left(\fracq\right)_4 \left(\fracq\right)_2 \ .

References

* * Theorems in algebraic number theory {{numtheory-stub

See also

References