# Cubic reciprocity

In mathematics, **cubic reciprocity** refers to various results connecting the solvability of two related cubic equations in modular arithmetic. It is a generalisation of the concept of quadratic reciprocity.

## 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

where and *a* and *b* are integers and

is a complex cube root of unity.

If is a prime element of *E* of norm *P* and is an element coprime to , we define the cubic residue symbol to be the cube root of unity (power of ) satisfying

We further define a *primary* prime to be one which is congruent to -1 modulo 3. Then for distinct primary primes and the law of cubic reciprocity is simply

with the supplementary laws for the units and for the prime of norm 3 that if then

## References

- David A. Cox,
*Primes of the form*, 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.