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}.

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