Degree and Basis

Question

I'm trying to express the field $\mathbb{Q}(\sqrt{3}, \sqrt[4]{3})$ as a simple extension over $\mathbb{Q}$. I know that $\left[ \mathbb{Q} (\sqrt{3}):\mathbb{Q} \right] =2 $ and $\left[ \mathbb{Q} (\sqrt[4]{3}):\mathbb{Q} \right] =4 $. My intuition tells me that $\left[ \mathbb{Q} (\sqrt{3}, \sqrt[4]{3}):\mathbb{Q} \right] =8 $, however when I work out basis, it seems like it should be 4. Where am I going wrong?

Answer 1

$\mathbb{Q}(\sqrt{3},\sqrt[4]{3})=\mathbb{Q}(\sqrt[4]{3})$ because $\sqrt{3}$ is already in $\mathbb{Q}(\sqrt[4]{3})$, as you've noticed. (This expresses the field as a simple extension over $\mathbb{Q}$.) Thus, the correct degree is indeed $$[\mathbb{Q}(\sqrt{3},\sqrt[4]{3}):\mathbb{Q}]=[\mathbb{Q}(\sqrt[4]{3}):\mathbb{Q}]=4$$ with an easy choice of basis being $\{1,\sqrt[4]{3},(\sqrt[4]{3})^2,(\sqrt[4]{3})^3\}$.