Find the dimension and a basis of the given extension field $\mathbb{Q} (\sqrt[3]{2}, \sqrt{3})$ of $\mathbb{Q}$.

Question

How do I find a base?

Find the dimension and a basis of the given extension field $\mathbb{Q} (\sqrt[3]{2}, \sqrt{3})$ of $\mathbb{Q}$.

Thanks in advance.

Answer 1

The fact that $\sqrt[3]2\notin \mathbb Q(\sqrt 3)$ is clear. Since $X^3-2$ is the minimal polynomial of $\sqrt[3]{2}$ on $\mathbb Q(\sqrt{3})$, you have $$[\mathbb Q(\sqrt[3]{2},\sqrt 3):\mathbb Q(\sqrt 3)]=3.$$ Also, $X^2-3$ is the minimal polynomial of $\sqrt 3$ over $\mathbb Q$, and thus

$$[\mathbb Q(\sqrt[3]2,\sqrt 3),\mathbb Q]=6.$$ A basis is given by $$\{1,\sqrt[3]{2},(\sqrt[3]{2})^2,\sqrt 3,\sqrt{3}\sqrt[3]2,\sqrt 3(\sqrt[3]2)^2\}$$