Analyze an argument about field extensions

Consider the following argument. (\omega is a cube root of 1.)

Because (\mathbb{Q}(\omega\sqrt[3]{2})/\mathbb{Q}) = (\mathbb{Q}(\sqrt[3]{2})) = 3, we have (\mathbb{Q}(\omega\sqrt[3]{2}/\mathbb{Q}(\sqrt[3]{2})) = 1, and thus \mathbb{Q}(\omega\sqrt[3]{2}) = \mathbb{Q}(\sqrt[3]{2}).

Is this reasoning valid?


This argument would only hold if one of \mathbb{Q}(\omega\sqrt[3]{2}) and \mathbb{Q}(\sqrt[3]{2}) contains the other. A more transparent example is \mathbb{Q}(\sqrt{2}) and \mathbb{Q}(\sqrt{3}); again these are degree 2 extensions of the rationals, but neither contains the other, so they are not equal.

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