I was wondering if there's a connection between the minimal polynomial of an endomorphism and the minimal polynomial of an element in a field extension.
Maybe if, by considering an algebraic extention K(α)|K, there's a way of creating an endomorphism in K(α), viewed as a K-vector field, in such a way that it's minimal polynomial is the same as the minimal polynomial of α in K.
Consider the multiplication map $f : K(\alpha) \to K(\alpha)$ defined by $f(x) = \alpha x$.
Viewing $K(\alpha)$ as a vector space over $K$, $f$ is a linear map.
The minimal polynomial $m_f(X) \in K[X]$ of the $K$-linear map $f$ is the monic polynomial of lowest degree such that $m_f(f)$ is the zero linear map, i.e. $m_f(\alpha)x = 0$ for all $x \in K(\alpha)$. If you think about it, this condition is equivalent to the condition that $m_f(\alpha) = 0$ (take $x = 1$).
So $m_f(X)$, the linear algebra minimal polynomial, is the same as $m_\alpha(X)$, the field-theoretic minimal polynomial