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A matroid is $\text{GF}(q)$-regular if it is representable over all propersuperfields of the field $\text{GF}(q)$. We show that, for highly connectedmatroids having a large projective geometry over $\text{GF}(q)$ as a minor, theproperty of $\text{GF}(q)$-regularity is equivalent to representability overboth $\text{GF}(q^2)$ and $\text{GF}(q^t)$ for some odd integer $t \geq 3$. Wedo this by means of an exact structural description of all such matroids.

Matroids representable over fields with a common subfield