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I have been told that $$ Aut(\mathbb{Q}[\sqrt[3]{2}] / \mathbb{Q}) = \{1\}, $$? but I can't figure out why the identity function is the only automorphism.

I understand that this is the group of $\mathbb{Q}$-automorphisms that are constant on $\mathbb{Q}$. But why is there only one of these?

Couldn't you for example have a map that moved the $\sqrt[3]{2}$ around some how? My thought is that the problem is the fact that "moving" it around one would end up with complex numbers, but I am not sure how to formally look at this.

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Let $f$ be an automorphism, since $f(2)=2$, we have $f(\sqrt[3]2)^3=2$. But there's an only element $x\in\mathbb{Q}[\sqrt[3]2]$ such that $x^3=2$, and that element is $\sqrt[3]{2}$.

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