The problem asks whether, when we restrict algebraic matroids to those representable over a specific finite field such as the field with two elements, the resulting class admits a computable and complete first-order axiomatization whose membership can be algorithmically verified. The recent undecidability result for recognizing algebraic matroids in general leaves open whether restricting the underlying field to a finite or otherwise tame algebraic structure rescues decidability. This is a natural and precise refinement: perhaps the full generality of algebraically closed fields of arbitrary characteristic is what introduces undecidability, while fixing the field collapses the complexity.