 Solution

Let $f$ be the common minimal polynomial. Evaluation givesso composing the two isomorphisms gives $\mathbb{Q}(\alpha )\cong \mathbb{Q}(\beta )$ and sends $\alpha$ to $\beta$.

Take $\alpha =\beta =\sqrt[4]{2}$. The field is real and contains the two roots $\pm \alpha$ of $X^4-2$, but not $\pm i\alpha$. Hence exactly two $\mathbb{Q}$-automorphisms are possible, while the degree is four.

In general, after choosing one isomorphism $\varphi :\mathbb{Q}(\alpha )\to \mathbb{Q}(\beta )$, every other one is uniquely $\varphi \circ \sigma$ for a $\mathbb{Q}$-automorphism $\sigma$ of $\mathbb{Q}(\alpha )$. ThusBy the automorphism-count divisibility theorem, this order divides $[\mathbb{Q}(\alpha ):\mathbb{Q}]$. Therefore $k$ always divides the degree.