I have been working on the following question:
Suppose that $K$ is an extension of $\Bbb{Q}$ of degree $n$. Let $\sigma_1,\dots,\sigma_n:K\hookrightarrow\Bbb{C}$ be the distinct embeddings of $K$ into $\Bbb{C}$. Let $\alpha\in K$. Regarding $K$ as a vector space over $\Bbb{Q}$, let $\phi:K\to K$ be the linear transformation $\phi(x) = \alpha x$. Show that the eigenvalues of $\phi$ are $\sigma_1(\alpha),\dots,\sigma_n(\alpha)$.
I don't quite understand what exactly the question is asking. I know that $\phi$ is an endomorphism of $K$, but $\sigma_i$ maps $K$ into $\Bbb{C}$. How can eigenvalues of $\phi$ be in $\Bbb{C}$?
Perhaps the question is considering maps $\phi_i:\sigma_i(K)\to\sigma_i(K)$ by $x\mapsto \sigma_i(\alpha)x$, in which case the answer seems fairly straightforward? Of course $\sigma_i(\alpha)$ is the eigenvalue for this map, just by definition.
The written answer I received, however, is far lengthier than this, and unfortunately is not very clear. Thus the above does not seem likely.
Any clarification or hints would be appreciated.