Is $e^{\pm 2\pi i/3}=1$ in the splitting field of $x^3-t\in \mathbb{F}_3(t)[x]$?

Question

Let $F=\mathbb{F}_3(t)$, where $t$ is a variable.

Then if $a$ is a root of $x^3-t\in F[x]$ in its splitting field we have $$ x^3-t=(x-a)^3=(x-\omega a)^3. $$ where $\omega$ is an abstract 3rd root of unity.

Q: Does this mean that $\omega=1$ in the splitting field of $x^3-t$?

Answer 1

The splitting field of $x^3-t$ is $\Bbb F_3(a)$. This is a normal but inseparable extension of $\Bbb F_3(t)$ with degree $3$.

In every field of characteristic $3$, there is only one cube root of unity, $1$ itself.

Answer 2

In any field of characteristic $p>0$, the only $p$th root of unity is $1$. Indeed, if $\omega$ is a $p$th root of unity, then $\omega^p-1=0$. But $\omega^p-1=(\omega-1)^p$, so $\omega-1=0$.