Example. [2106C]

$F = \mathbb{Q}$ and $K = \mathbb{Q}(\sqrt[3]{ 2 })$. The minimal polynomial of $\sqrt[3]{ 2 }$ over $\mathbb{Q}$ is $x^{3} - 2$. $x^{3} - 2$ has three roots $$ \theta_{1} = \sqrt[3]{ 2 }, \quad \theta_{2} = \sqrt[3]{ 2 }e^{ 2 \pi i / 3 } = \sqrt[3]{ 2 } \frac{-1 + \sqrt{ 3 }}{2}, \quad \theta_{3} = \sqrt[3]{ 2 } e^{ 4 \pi i / 3 } = -\frac{\sqrt[3]{ 2 }(1 + \sqrt{ 3 })}{2} $$ in $\overline{\mathbb{Q}}$, but only $\sqrt[3]{ 2 }$ belongs to $\mathbb{Q}(\sqrt[3]{ 2 })$. $x^{3} - 2$ has only one root in $\mathbb{Q}(\sqrt[3]{ 2 })$. By Example 2106 (2), $$ \mathrm{Aut}\left( \mathbb{Q}(\sqrt[3]{ 2 }) / \mathbb{Q} \right) = \{ 1 \}, \qquad \text{the trivial group}. $$