  We develop the theory of matroids over one-dimensional algebraic groups, with
special emphasis on positive characteristic. In particular, we compute the
Lindstr\"om valuations and Frobenius flocks of such matroids. Building on work
by Evans and Hrushovski, we show that the class of algebraic matroids, paired
with their Lindstr\"om valuations, is not closed under duality of valuated
matroids.
