Find all solutions to the equation $\displaystyle x^{16} = 1$ in the field $\displaystyle Z_{17}$.
A solution or a starting point would be greatly appreciated.
from $\displaystyle x^{-1}=1$ we have $\displaystyle x=1$ because the only element whose inverse is 1 is itself 1 . Moreover since the order of each element divides the order of the group and the order of this group is 16 then $\displaystyle x^{16}=1$ and $\displaystyle x^{16}=x^{17}x^{-1}$ but $\displaystyle x^{17}=x$ and so $\displaystyle x^{16}=x.x^{-1}=1$ and as you see it won't be $\displaystyle x^{-1}$.