I am taking a course in number theory that deals with quadratic and (higher order) reciprocity and have been wondering about the following question: Fermat's last theorem states that there are no integer solutions to $x^n+y^n=z^n$ for $n>2$. Does a similar result exists over finite fields?
My first thought was looking$\mod p$ and then for $n$ such that $\gcd(n,p-1)=1$ the map $$\phi :F_p \rightarrow F_p$$ $$\phi(a)=a^n$$ is an automorphism and therefore a solution exists.
Also, for every finite field we could reduce the equation to $$1+y^n=z^n$$ by multiplying by $(x^{-1})^{n}$.
I can't seem to get much further and would be glad to hear about existing results on the subject.
Answer
Schur proved that for every $n$, if $p$ is a large enough prime, then there is a nontrivial solution to $x^n+y^n\equiv z^n\bmod p$. See this link.
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