We define an algebraic extension $K/F$ to be normal if every irreducible $f \in F[x]$ with one root in $K$ splits in $K[x]$.
Now, in my lectures it was stated that $\mathbb{Q}(i)$ is a normal extension of $\mathbb{Q}$. Why is this the case?
If one has an irreducible $f \in \mathbb{Q}[x]$ with a root $\alpha \in \mathbb{Q}(i)$, then why does this imply that $f$ has all its roots in $\mathbb{Q}(i)$?
Certainly we have $\alpha^*$ (the complex conjugate of $\alpha$) is also a root of $f$ in $\mathbb{Q}(i)$, but how do we know we have all the roots of $f$ in $\mathbb{Q}(i)$?