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 xn+yn=zn 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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