Prove that $\mathbb{Q}(\sqrt[4]{2}, i)$ is a splitting field over $\mathbb{Q}$.
I know basic Group, Ring, and Field theory, and I've read the definition of a splitting field, yet I still have no idea where to start on this one.
Is it asking me to show that any polynomial in $\mathbb{Q}$ will split over $\mathbb{Q}(\sqrt[4]{2}, i)$?
It is the splitting field of $X^4-2$. Let $F$ be the splitting field of $X^4-2$. It contains $\sqrt[4]2$ and $i\sqrt[4]2$ which are roots of $X^4-2$. this implies that it contains $(\sqrt[4]2)^3$ and $i\sqrt[4]2(\sqrt[4]2)^3=2i$, so it contains $Q(\sqrt[4]2,i)$. Since $Q(\sqrt[4]2,i)$ contains all the roots of $X^4-2$, it is its splitting field.