Quartic reciprocity

Remarks

The proof follows the same pattern as for cubic reciprocity, with quartic Gauss sums and Jacobi sums over the field with $p$ elements for $p \equiv 1 \pmod 4$. Each prime of $\Z[i]$ not dividing $1 + i$ has exactly one primary associate. The sign is $-1$ only when both $\lambda$ and $\pi$ are congruent to $3 + 2i$ modulo $4$. A classical consequence: for a prime $p \equiv 1 \pmod 8$, the integer $2$ is a fourth power modulo $p$ if and only if $p = x^2 + 64y^2$ for some integers $x, y$. For example, $73 = 3^2 + 64 \cdot 1^2$, and $2^{18} \equiv 1 \pmod{73}$, so $2$ is a fourth power modulo $73$. The prime $17$ is not of this form, and the nonzero fourth powers modulo $17$ are $1$, $4$, $13$ and $16$.

Going deeper

For a student

With supplementary laws for $i$ and $1 + i$, the law lets one compute any quartic residue character by swapping and reducing. It answers questions such as when $2$ is a fourth power modulo a prime, which quadratic reciprocity cannot decide. Like cubic reciprocity, it is a special case of the Artin reciprocity law.