Cubic reciprocity

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

1 Algebraic setting
2 Primary: the definition
3 References
4 External links

[edit] 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 $π$ is a prime element of E of norm P and $α$ is an element coprime to $π$ , we define the cubic residue symbol $\left(\frac{\alpha}{\pi}\right)_3$ to be the cube root of unity (power of $ω$ ) 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 $π$ and $θ$ 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 $1 - ω$ of norm 3 that if $π = - 1 + 3(m + n ω)$ then

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

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

[edit] Primary: the definition

The definition here of primary is very traditional, going back to the original papers of Ferdinand Eisenstein. The presence of the minus sign is not so easily compatible with modern definitions, for example in discussing the conductor of a Hecke character. It is in any case straighforward to move the minus sign elsewhere, given that the cubic character of −1 is not in doubt.

[edit] References

David A. Cox, Primes of the form $x 2 + n y 2$ , 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.