The normal closure of a field extension I'm making my first steps in abstract algebra and I was wondering, if there is a technique to determine the normal closure of a given extension, cause all I know is a theoretical definition: $K_n$ is a normal closure of an extension $K/k$ if it's the intersection of all the normal extensions that contain $K$. 

As an application, what is the normal closure of $\mathbb Q(i,j\sqrt[3]{2})$? 

thank you for your time

All normal extensions are splitting fields and vice versa. 
In your particular case you have two algebraic numbers, that is, $i$ and $j\sqrt[3]{2}$. The spliting field of the first is $\mathbb Q(i)$, while of the second is $\mathbb Q(j, \sqrt[3]{2})$. Now take their compositum and this is the normal closure. 
