Solve $ x^2 = 2$ over $ F_5 $.

Question

Since $ F_5 $ is isomorphic to $ \Bbb Z_5 $, I tried to solve this equation over $ \Bbb Z_5 $.

Since $ gcd(2,5)=1 $, $ \Bbb Z_5 $ contains a primitive $2$nd root of unity.

So if $ \omega $ is the primitive root of unity and $ \alpha $ is any root, then the roots would be $ \alpha $ and $\alpha \omega $.

Is this approach correct or is there some other way to solve this$?$

Answer 1

Since $F_5$ is a field with only five elements, it is perhaps simplest to solve the equation by just trying each element.

Answer 2

$x^2=x\iff x^2-x=x(x-1)=0$ Since it is a quadratic equation a coefficients in a field, there are just two solutions which are given by the last equality, (i. e. $1$ and $0$)