An after-dinner mint

Posted on May 25, 2014 by arithmetica

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.

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