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

$z=a+b\,\omega$

where and a and b are integers and

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

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 $\left({\frac {\alpha }{\pi }}\right)_{3}$ to be the cube root of unity (power of $\omega$) satisfying

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

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

$\left({\frac {\pi }{\theta }}\right)_{3}=\left({\frac {\theta }{\pi }}\right)_{3}$

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

$\left({\frac {\omega }{\pi }}\right)_{3}=\omega ^{m+n}$

$\left({\frac {1-\omega }{\pi }}\right)_{3}=\omega ^{2m}$

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.

External links

• Cubic Reciprocity Theorem from MathWorld