  We show that a field extension $K\subseteq L$ in positive characteristic $p$
and elements $x_e\in L$ for $e\in E$ gives rise to a matroid $M^\sigma$ on
ground set $E$ with coefficients in a certain skew hyperfield $L^\sigma$. This
skew hyperfield $L^\sigma$ is defined in terms of $L$ and its Frobenius action
$\sigma:x\mapsto x^p$. The matroid underlying $M^\sigma$ describes the
algebraic dependencies over $K$ among the $x_e\in L$ , and $M^\sigma$ itself
comprises, for each $m\in \mathbb{Z}^E$, the space of $K$-derivations of
$K\left(x_e^{p^{m_e}}: e\in E\right)$.
  The theory of matroid representation over hyperfields was developed by Baker
and Bowler for commutative hyperfields. We partially extend their theory to
skew hyperfields. To prove the duality theorems we need, we use a new axiom
scheme in terms of quasi-Pl\"ucker coordinates.
