Jacobi Sums and Fermat’s theorem on Sums of Squares.

Look at the Gauss Sum

$\displaystyle G(\chi) = \sum_{ x \mod p} \chi(x) e(\frac{2\pi ix}{p})$

for a multiplicative character ${\mod p.}$

It’s easy to see by executing the double sum ${G(\chi) \bar G(\chi)}$ (with a change of variables) that

$\displaystyle |G(\chi)| =\sqrt{p},$

(determining sign is more harder- you can apply poisson summation or Gauss’s ${q}$ -binomial identities etc to determine the sign)

Consider the Jacobi Sums.

$\displaystyle J(\chi, \psi) = \sum_{x \mod p} \chi(x) \psi(1-x)$

Gauss Sums are the analogues of the Gamma function and Jacobi Sums are analogue of Beta function.

We have the following identity that relates Jacobi Sums to Gauss Sum- the analogue to the identity ${B (a, b) =\frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}}$

$\displaystyle J(\chi, \psi) =\frac{G(\chi)G(\psi)}{G(\chi\psi)}$

provided ${\chi\psi}$ is not a trivial character.

It is easy to see the above equality starting from ${G(\chi) G(\psi)}$ which is a sum over ${x_1, x_2 \mod p}$ and a change of variables ${ x_2 \rightarrow x_1t}$

Fermat’s theorem: We want to prove that every prime ${p \equiv 1 \mod 4}$ is a sum of squares, that is there are integers ${a, b}$ such that ${p =a^2+b^2.}$