Example of a nontransitive action of $\operatorname{Aut}(K/\mathbb Q)$ on the roots in $K$ of an irreducible polynomial.

Solution 1:

Take $f(x)=x^2-2$, $K=\mathbb{Q}(\sqrt[4]{2})$. Then $K$ contains two roots of $f$, but there is no automorphism exchanging them because only one of the roots is a square in $K$.

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Example of a nontransitive action of $\operatorname{Aut}(K/\mathbb Q)$ on the roots in $K$ of an irreducible polynomial.

Solution 1:

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