Is $(\mathbb{Q} (\sqrt[3]{2}), +, \cdot)$ a field ? I came across this question while doing homework, but it was unlike any other problem I faced. For instance, how would one show that : $(a+b\sqrt[3]{2})\cdot(c+d \sqrt[3]{2})$ belongs to $\mathbb{Q} (\sqrt[3]{2})$.
Thank you
$\mathbb{Q}(\sqrt[3]{2})$ is by definition, the smallest field containing $\mathbb{Q}$ and $\sqrt[3]{2}$.
Thus, there is nothing to prove.
The fact that $\mathbb{Q}[\sqrt[3]{2}]$ is a field may need further proof. Note the difference between round bracket and square bracket!