## Show that a given set is linearly independent over the rationals

Prove that $\{1, \sqrt[3]{2}, \sqrt[3]{4} \}$ is linearly independent over $\mathbb{Q}$.

Note that $p(x) = x^3 - 2$ is irreducible over $\mathbb{Q}$ by Eisenstein’s criterion, and moreover $\sqrt[3]{2}$ is a root of $p(x)$. As we saw in Theorem 4.6 in TAN, every element in $\mathbb{Q}(\sqrt[3]{2})$ is uniquely of the form $a + b \sqrt[3]{2} + c\sqrt[3]{2}$. In particular, 0 is uniquely of this form. Hence $\{1,\sqrt[3]{2}, \sqrt[3]{4}\}$ is linearly independent over $\mathbb{Q}$.