Find and show primitive element of field extension

Question

I have the polynomial $\ f(x)= x^{15}-3 $ and want to find the splitting field. The splitting field is surely $\Bbb Q$ adjoined the roots of the polynomial. I want to write the splitting field as $\Bbb Q$ adjoined one element. How should I choose this element? How do I show that $\Bbb Q$ adjoined this element gives me the whole splitting field?