# Finite fields containing cube root of unity and square root of $-3$

Let $q=p^n$.

1. For what $q$ does $\mathbb F_q$ contain a primitive cube root of unity?
2. Deduce for which $q$ the polynomial $x^2+x+1$ splits into linear factors in $\mathbb F_q[x]$. Use the quadratic formula when appropriate to find these factors. Over which fields is it inappropriate to use the quadratic formula?
3. Which finite fields contain a square root of $-3$ and which do not?

If $x$ is a primitive cube root of unity in $\mathbb F_q$, then $x\ne 1, x^2\ne 1$ but $x^3=1$ in $\mathbb F_q$. Equivalently, $x^3-1=0$ or $(x-1)(x^2+x+1)=0$. Since $x$ is primitive and $\mathbb F_q$ has no zero divisors, $x^2+x+1=0$. Now I believe the fields containing a primitive cube root of 1 are the fields over which this polynomial splits. So are the first two parts asking the same question? Anyway, I don't know how to describe the fields having either property.

For the quadratic formula part, $x=\frac{-1\pm \sqrt{-3}}{2}$; this is valid provided $2\ne 0$ i.e. provided the characteristic isn't 2. So the polynomial splits iff $\sqrt{-3}$ lies in the field (provided the characteristic isn't 2). So is the third part asking the same as the first two?

Let $F=\mathbb F_q$.