Definitions

# Cubic reciprocity

In mathematics, cubic reciprocity is any of various results connecting the solvability of two related cubic equations in modular arithmetic.

## Algebraic setting

The law of cubic reciprocity is most naturally expressed in terms of the Eisenstein integers, that is, the ring E of complex numbers of the form

$z = a + b,omega$

where and a and b are integers and

$omega = frac\left\{1\right\}\left\{2\right\}\left(-1 + isqrt 3\right) = e^\left\{2pi i/3\right\}$

is a complex cube root of unity.

If $pi$ is a prime element of E of norm P and $alpha$ is an element coprime to $pi$, we define the cubic residue symbol $left\left(frac\left\{alpha\right\}\left\{pi\right\}right\right)_3$ to be the cube root of unity (power of $omega$) satisfying

$left\left(frac\left\{alpha\right\}\left\{pi\right\}right\right)_3 equiv alpha^\left\{\left(P-1\right)/3\right\} mod pi.$

We further define a primary prime to be one which is congruent to -1 modulo 3, still in the ring E; since any prime will still be prime when multiplied by a unit of the ring E, a sixth root of unity, this is not a deep restriction. Then for distinct primary primes $pi$ and $theta$ the law of cubic reciprocity is simply

$left\left(frac\left\{pi\right\}\left\{theta\right\}right\right)_3 = left\left(frac\left\{theta\right\}\left\{pi\right\}right\right)_3$

with the supplementary laws for the units and for the prime $1-omega$ of norm 3 that if $pi = -1 + 3\left(m+nomega\right)$ then

$left\left(frac\left\{omega\right\}\left\{pi\right\}right\right)_3 = omega^\left\{m+n\right\}$
$left\left(frac\left\{1-omega\right\}\left\{pi\right\}right\right)_3 = omega^\left\{2m\right\}.$

Since

$left\left(frac\left\{thetaphi\right\}\left\{pi\right\}right\right)_3 = left\left(frac\left\{theta\right\}\left\{pi\right\}right\right)_3 left\left(frac\left\{phi\right\}\left\{pi\right\}right\right)_3$
the cubic residue of any number can be found once it is factored into primes and units.

## Note on the definition of "primary"

The definition here of primary is the traditional one, going back to the original papers of Ferdinand Eisenstein. The presence of the minus sign is not easily compatible with modern definitions, for example in discussing the conductor of a Hecke character. But if so desired, it is straightforward to move the minus sign elsewhere, as −1 is a cube, in fact, the cube of −1.

## References

• David A. Cox, Primes of the form $x^2+ny^2$, Wiley, 1989, ISBN 0-471-50654-0.
• K. Ireland and M. Rosen, A classical introduction to modern number theory, 2nd ed, Graduate Texts in Mathematics 84, Springer-Verlag, 1990.
• Franz Lemmermeyer, Reciprocity laws: From Euler to Eisenstein, Springer Verlag, 2000, ISBN 3-540-66957-4.