= Solution
Differentiate the components directly:
$$
\begin{aligned}
(\nabla\times\mathbf F)_x&=\partial_y(-3xz^2-1)-\partial_z(x^2-2y)=0,\\
(\nabla\times\mathbf F)_y&=\partial_z(2xy-z^3)-\partial_x(-3xz^2-1)=-3z^2+3z^2=0,\\
(\nabla\times\mathbf F)_z&=\partial_x(x^2-2y)-\partial_y(2xy-z^3)=2x-2x=0.
\end{aligned}
$$
Thus $\boxed{\nabla\times\mathbf F=0}$. The field is smooth on all of $\mathbb R^3$, a <simply connected domain>, so the zero-curl condition is sufficient for a <conservative vector field>. The domain condition matters: zero <curl> alone need not imply a global potential on a region with holes.
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