# Gauss sums

Friday, March 27, 2015 - 9:10am - 9:40am

Edray Goins (Purdue University)

In 1798, Carl Friedrich Gauss counted the number of solutions modulo various primes $p$ of the equation $a^3 + b^3 + c^3 = 0$, and found a surprising connection with the ability to write the prime in the form \, p = a_p^2 + 27 \, b_p^2$ for some integers $a_p$ and $b_p$. In 1846, Ernst Eduard Kummer gave a careful study of Gauss' proof using the cubic Gauss sums $\tau_p = \sum_{\alpha \in \mathbb F_p} {\zeta_p}^{\alpha^3}$.