Cubic reciprocity

Remarks

The standard proof uses cubic Gauss sums and Jacobi sums. For a prime $\pi$ of norm $p \equiv 1 \pmod 3$, the Jacobi sum of the cubic character is a primary element of norm $p$. The cube of the Gauss sum equals $p$ times this Jacobi sum. Comparing this identity modulo the other prime gives the law. Each prime of $\Z[\omega]$ not dividing $3$ has exactly one primary associate, so the hypothesis "primary" only fixes the choice among the six associates. A classical consequence: for a prime $p \equiv 1 \pmod 3$, the integer $2$ is a cube modulo $p$ if and only if $p = x^2 + 27y^2$ for some integers $x, y$. For example, $31 = 2^2 + 27 \cdot 1^2$, and indeed $4^3 = 64 \equiv 2 \pmod{31}$. The prime $7$ is not of this form, and the cubes modulo $7$ are only $0$, $1$ and $6$.

Going deeper

For a student

The law is usually completed by supplementary laws for the units and for the prime $1 - \omega$ above $3$. Together they let one compute any cubic residue character by repeated swapping and reduction, just as with the Legendre symbol. The proof through Gauss and Jacobi sums is the model for the higher reciprocity laws, and the general case is part of class field theory through the Artin reciprocity law.