roots of unity in finite fields from complex roots of unity

Question

After watching this video containing some examples of roots of unity in finite fields obtained from complex roots of unity, I became curious about whether there is a general theorem behind it. For example, it looks it might imply that, provided 3 divides $p-1$ where $p$ is a large enough prime, there is a cube root of unity $x$ mod $p$ such that $4x^2 \equiv -3 \pmod p$. Can anybody help me out here? (I posted a comment under the video asking this question but the official response was, as you can see there, "dunno".)

Answer 1

I won't waste my time looking at a video, but this is easy to reverse engineer. CWing this because we are very likely to have covered this already on our site (but I don't have the time to look for dupe now).

• A third root of unity, in any field $F$, is a solution of the equation $x^3-1=0$.
• The factorization $x^3-1=(x-1)(x^2+x+1)$ is true over any field. When we disallow $1$ as a third root of unity (wanting a primitive third root of unity), it follows that a third root of unity must be a solution of the quadratic $$x^2+x+1=0.$$
• If $1_F+1_F\neq 0_F$ (equivalently: we are not in characteristic two), the usual quadratic formula can be proved by the usual argument of completing the square, and tells us that the solutions of $x^2+x+1=0$ are $$x=-\frac12\pm\sqrt{\frac{-3}4}.$$
• Therefore there are primitive roots of unity in a field $F$ if and only if the element $-3/4$ has a square root in $F$.
• Because we know that the multiplicative group of the field $F=\Bbb{Z}/p\Bbb{Z}$ is cyclic of order $p-1$, it follows that there is a primitive third root of unity in $F$ if and only if $3\mid p-1$.
• On the other hand the element $-3/4$ has a square root $x\in \Bbb{Z}/p\Bbb{Z}$ if and only if $x$ is a solution of the congruence $$4x^2\equiv-3\pmod p.$$

Putting the three last bullets together is the explanation.