I am studying Galois theory, and when talking about field extensions the eigenvalues of an element come up. For $L/K$ a finite field extension and $x \in L$ we consider $[ \times x] \in \operatorname{End}(L)$ defined by $y \longrightarrow xy$. Then we start to do a bunch of linear algebra, and everything is fine until we consider the eigenvalues. At that point, I can still follow the linear algebra that is done but there is something I really can't get my head around.
Say that $\lambda \in K$ is an eigenvalue of $x$ and $y \in L^\times$ is an eigenvector. Then we get by definition $xy = \lambda y$. Since all elements are in $L$ we can divide both sides by $y$ and get $x =\lambda$. So every $x \in K$ has only itself as an eigenvalue and every $x \in L \, \backslash \, K$ has no eigenvalues at all.
Clearly there is something wrong in this reasoning but I cannot understand what it is... Thanks in advance.