Minimal polynomial of $\alpha^2$ over a field if $\alpha^3+\alpha-1=0$

Question

In a certain exercise I am asked to find the minimal polynomial over a field $K$ of the element $\alpha^2$, where $\alpha$ is such that $p(\alpha) = 0$, where $p(x) = x^3 + x - 1$ and it is the minimal polynomial (edited).
I found a polynomial that is satisfied by $\alpha^2$ with coefficients in $K$, taking into account that $\alpha = \alpha^2 + \alpha^4$, and since $p(\alpha)^2 = 0^2 = 0$, then $(\alpha^2)^3+ 2(\alpha^2)^2 + \alpha^2 - 1 = 0$, then the minimal polynomial is $q(x) = x^3 + 2x^2 + x -1$. The problem here is that I don't know if it is irreducible. If I knew that the field is $\mathbb{Q}$, then it would be straightforward, but I don't know if is reducible or not in an unspecified field $K$. I would need to know if it has roots, but I don't know how. Can somedoby help me? Btw, is the polynomial right?