## An after-dinner mint

Suppose $p$ is a prime congruent to 1 mod 3, so the equation $x^2 = -3 \;\mathrm{mod}\; p$ has a solution. How do you find $x$ explicitly? One answer: choose $u \in (\mathbf{Z}/p)^{\times}$ of order 3, and set $x = 2u+1$.  There is a much more general version of this, which can be explained using finitary Gauss sums.