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Let M be a matroid representable over a (partial) field P and B a matrixrepresentable over a sub-partial field P' of P. We say that B confines M to P'if, whenever a P-representation matrix A of M has a submatrix B, A is a scaledP'-matrix. We show that, under some conditions on the partial fields, on M, andon B, verifying whether B confines M to P' amounts to a finite check. Acorollary of this result is Whittle's Stabilizer Theorem.  A combination of the Confinement Theorem and the Lift Theorem fromarXiv:0804.3263 leads to a short proof of Whittle's characterization of thematroids representable over GF(3) and other fields.  We also use a combination of the Confinement Theorem and the Lift Theorem toprove a characterization, in terms of representability over partial fields, ofthe 3-connected matroids that have k inequivalent representations over GF(5),for k = 1, ..., 6.  Additionally we give, for a fixed matroid M, an algebraic construction of apartial field P_M and a representation A over P_M such that everyrepresentation of M over a partial field P is equal to f(A) for somehomomorphism f:P_M->P. Using the Confinement Theorem we prove an algebraicanalog of the theory of free expansions by Geelen et al.

Confinement of matroid representations to subsets of partial fields