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I was asked to prove the galois group of a given normal extension is non-abelian. My original solution was to use isomorphism extension theorem but that was not taught in class. So, in my new attempt to solve it, I hit a road bum.

To show non-abelian, I need 2 automorphisms. I already know the 2 automorphisms exist but I don't know how to prove one of them exist.

Let $K$ be a normal extension of $\mathbb{Q(\sqrt[3]7)}$. How would I prove that given any element $a\in K$ and $b$, a conjugate of $a$, there exist an automorphism $\alpha\in Gal(K/\mathbb{Q})$ such that $\alpha(a)=b$? Let's assume $K$ is a finite dimension over $\mathbb{Q(\sqrt[3]7)}$ because infinite dimension was not taught in class.


EDIt: Added a few more details

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