Observe $f=x^3-5$ over $\mathbb{Z}_{11}$. So we can write $f=(x-3)(x^2+3x+9)$. Let $\alpha$ be a root of $x^2+3x+9$. How do we find another root?
This problem is related to finding a splitting field extension. I suppose that I should get that 2nd root is $\alpha$ times something and in that way I would have that $\mathbb{Z}_{11}(\alpha)$ is desired splitting field, but I don't know how to find another root. How can it be done?
What is the general approach to the problems like this?
as you already have the root $3,$ let $\omega$ be a root of $$t^2 + t + 1$$ and thus a new cube root of $1.$ The cube roots of $5$ are now $3, 3 \omega, 3 \omega^2.$
Note that $t= \omega^2$ is also a root of $t^2 + t + 1,$ since $\omega^4 = \omega$