I am trying to express the splitting field of $x^4-2x^3+5x^2-2x+4$ as a radical extension of $\mathbb{Q}$.
I found the roots of the above polynomial and found that the splitting field the above polynomial is $K=\mathbb{Q}(\sqrt{3}, i)$. Hence I can form a tower $\mathbb{Q}\subset\mathbb{Q}(\sqrt{3})\subset\mathbb{Q}(\sqrt{3}, i)$ and $\sqrt{3}^2, i^2\in \mathbb{Q}$. Is my approach correct?