I need to answer a question "Does $x^4 \equiv -17 \pmod{83}$ has root or not?" Here is my answer.
We first prove that $X^2 \equiv -17 \pmod{83}$ has no root by using Legendre symbol. Indeed, $\left( \dfrac{-17}{83} \right) =\left(\dfrac{-1}{83} \right) \left( \dfrac{17}{83} \right)=-\left(\dfrac{17}{83} \right) =-\left( \dfrac{83}{17}\right) $
$=-\left(\dfrac{15}{17} \right) = -\left( \dfrac{3}{17}\right) \left( \dfrac{5}{17} \right) =-\left( \dfrac{17}{3}\right) \left( \dfrac{17}{5}\right) =-\left( \dfrac{2}{3}\right) \left(\dfrac{2}{5} \right)=-1 $
So it is clear that $X^2 \equiv -17 \pmod{83}$.
It follows readily from here that $x^4 \equiv -17 \pmod{83}$ has not root.
Is it a correct answer? My teacher told me my answer is incorrect and ask me to reflect it. Honestly, after sober consideration, i can't find any mistake on this solution. Any help is appricated.