Find a subfield of $\mathbb{C}$ isomorphic to other field

Question

Do you know, how I can find a subfield of $\mathbb{C}$ isomorphic to $F = \mathbb{Q}(\sqrt[3]{7})$ such that $F \nsubseteq \mathbb{R}$? I don't even have clue, how I should start.

Thanks

Answer 1

$\sqrt[3]{7}$ has minimal polynomial over $\mathbb{Q}$ given by $f:=x^{3}-7$. This polynomial has two complex roots, say $\alpha$ and $\beta$ (find them!), which do not belong to $\mathbb{R}$. Since each extension of $\mathbb{Q}$ given by adjoining a root of $f$ to $\mathbb{Q}$ is isomorphic to $\mathbb{Q}[x]/(x^{3}-7)$, $F$ is isomorphic both to $G:=\mathbb{Q}(\alpha)$ and to $H:=\mathbb{Q}(\beta)$. Since both $G$ and $H$ are not contained in $\mathbb{R}$, you are done.