text: Q:
  
  Fermat's Last Theorem in finite fields
  
  Consider the finite field $\mathbb{F}_q$. Schur (1916) proved that, given $n$, when the field is sufficient large, this equation,
  $$x^n+y^n= z^n$$
  always has a nontrivial solution.
  What conditions does the number of solutions satisfy?
  
  I. Schur, Über die Kongruenz $x^{m} + y^{m} \equiv z^{m} \pmod{p}$, Jahresber. Deutschen Math. Verein. 25 (1916), 114–117.
  
  A:
  
  There are some results and references in Lang, Cyclotomic Fields, 1.§6 (p. 22ff. in Cyclotomic Fields I and II, Combined Second Edition).
  Let $V(d)$ be the Fermat curve of degree $d$. Theorem 6.1. The number of points of $V(d)$ (in affine space) is $q^2 - (q-1)\sum\chi^{a+b}(-1)J(\chi^a,\chi^b)$, the sum over integers $a,b$ with $0 < a,b < d$ and $a+b \not\equiv 0 \pmod{d}$ and $\chi$ the character such that $\chi(u) = \omega(u)^{(q-1)/d}$ with $\omega: \mathbf{F}_q \to \mu_{q-1}$ the Teichmüller character and the Jacobi sum $J(\chi_1,\chi_2) = -\frac{S(\chi_1)S(\chi_2)}{S(\chi_1\chi_2)}$ and the Gauß sum $S(\chi) = \sum_u\chi(u)\lambda(u)$ and $\lambda: \mathbf{F}_q \to \mu_p, \lambda(x) = \exp(2\pi i/p\mathrm{Tr}(x))$.
  
  A:
  
  It is more or less easy to obtain, via exponential sums, an asymptotic formula for the number $J$ of solutions of the congruence 
  $$x^{n}+y^{n} \equiv z^{n} \pmod{p}$$
  where $$1 \leq x, y, z \leq p-1.$$
  Indeed, since
  \begin{eqnarray*} J &=& \sum_{x=1}^{p-1} \sum_{y=1}^{p-1} \sum_{z=1}^{p-1} \frac{1}{p}\sum_{a=0}^{p-1}e^{2 \pi i  \frac{a(x^{n}+y^{n}-z^{n})}{p}}\\
  &=& \frac{(p-1)^{3}}{p}+ \frac{1}{p} \sum_{a=1}^{p-1} \sum_{x=1}^{p-1}\sum_{y=1}^{p-1} \sum_{z=1}^{p-1}e^{2\pi i \frac{a(x^{n}+y^{n}-z^{n})}{p}}\\
  &=& \frac{(p-1)^{3}}{p} + \frac{1}{p}\sum_{a=1}^{p-1}\left|\sum_{x=1}^{p-1}e^{2\pi i \frac{ax^{n}}{p}}\right|^{2}\sum_{y=1}^{p-1}e^{2\pi i \frac{ay^{n}}{p}}\\
  \end{eqnarray*}
  and
  $$ \sum_{a=0}^{p-1} \left| \sum_{x=1}^{p-1} e^{2\pi i \frac{ax^{n}}{p}} \right|^{2} \leq n\,p\,(p-1)$$
  and
  $$\left|\sum_{y=1}^{p-1}e^{2\pi i \frac{\alpha y^{n}}{p}}\right| \leq n\sqrt{p}$$
  for any integer $\alpha$ which is not divisible by $p$, it follows that
  $$J = \frac{(p-1)^{3}}{p}+O\left(n^{2}(p-1)\sqrt{p}\right).$$
  
  