How should I find Splitting Field of $x^3-2$ over $\mathbb Q$. How should I find Splitting Field of $x^3-2$ over  $\mathbb Q$.
**My try **:
$x^3-2=(x-2^\frac{1}{3})(x^2+2^\frac{1}{3}x+2^\frac{2}{3})$
On solving I am getting the roots as $2^\frac{1}{3},\dfrac{2^\frac{1}{3}[-1+\sqrt 3 i]}{2},\dfrac{2^\frac{1}{3}[-1-\sqrt 3 i]}{2},$
But I dont know the answer is given that roots are $2^\frac{1}{3},2^\frac{1}{3}\omega ,2^\frac{1}{3}\omega ^2$ where $\omega $ is cube root of unity

Splitting field will be $\mathbb{Q}(2^{1/3},\omega)$ where w is cube roots of unity. 
where $\omega =\dfrac{-1+\sqrt 3 i}{2}~\&~\omega ^2=\dfrac{-1-\sqrt 3 i}{2},$
