Abstract: The theory of matroid representation over hyperfields was developed 
by Baker and Bowler for commutative hyperfields. We partially extend their 
theory to skew hyperfields, using a new axiom scheme in terms of quasi-Plucker 
coordinates.

The immediate motivation for making this generalization comes from the study 
of algebraic matroids. An algebraic matroid arises from a finite set E, a field 
extension L/K, and an element x_e in L for each e in E. Then the algebraic 
matroid M(K,x) has ground set E, and a subset F of E is dependent in the matroid 
exactly if {x_e : e in F} is algebraically dependent over K.

Previous work of Bollen, Draisma and Pendavingh showed that if K has positive 
characteristic, then an algebraic representation (K,x) induces a matroid 
valuation of M(K,x), called the Lindström valuation. It was show by Cartwright 
that this valuation determines the inseparable index of subfields K(x_e: e in B) 
in L, for each basis B of M(K,x).

We show that (K,x) in fact gives a matroid over a certain skew hyperfield, 
which comprises more detailed information such as the the space of K-derivations 
of K(x_e: e in E), as well as the Lindström valuation.