Identifying the splitting field of $x^{4}-2x^{2}-3$

Question

Identify the splitting field F of $f(x)=x^{4}-2x^{2}-3$ over $\mathbb{Q}$ and determine $\alpha\in\mathbb{C}$ such that $F=\mathbb{Q}(\alpha)$.

My thoughts: Clearly $f(x)$ isn't irreducible, since $f(x)=(x^{2}+1)(x^{2}-3)$. So, the roots of $f(x)$ are $\pm i,\pm\sqrt{3}$. As far as I understand, I need to find an isomorphism between $\mathbb{Q}(i,\sqrt{3})$ and $\mathbb{Q}(\alpha)$. Could you please give me a hint to find such isomorphism?

Answer 1

$\mathbf Q(i,\sqrt3)=\mathbf Q(i+\sqrt3$).

Indeed $(i+\sqrt3)^3=8i$, so $i=\frac18(i+\sqrt 3)^3$ and $\sqrt3=(i+\sqrt 3)-\frac18(i+\sqrt 3)^3$.

Note it is generated by $i-\sqrt3$ as well. More generally you may try to obtain $i$ from $i+\lambda\sqrt 3$ for some $\lambda$.