  The notion of thin sums matroids was invented to extend the notion of
representability to non-finitary matroids. A matroid is tame if every
circuit-cocircuit intersection is finite. We prove that a tame matroid is a
thin sums matroid over a finite field k if and only if all its finite minors
are representable over k.
  We expect that the method we use to prove this will make it possible to lift
many theorems about finite matroids representable over a finite field to
theorems about tame thin sums matroids over these fields. We give three
examples of this: various characterisations of binary tame matroids and of
regular tame matroids, and unique representability of ternary tame matroids.
