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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".)

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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.

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