  It is proved that for each prime field $GF(p)$, there is an integer $f(p)$
such that a 4-connected matroid has at most $f(p)$ inequivalent representations
over $GF(p)$. We also prove a stronger theorem that obtains the same conclusion
for matroids satisfying a connectivity condition, intermediate between
3-connectivity and 4-connectivity that we term "$k$-coherence".
  We obtain a variety of other results on inequivalent representations
including the following curious one. For a prime power $q$, let ${\mathcal
R}(q)$ denote the set of matroids representable over all fields with at least
$q$ elements. Then there are infinitely many Mersenne primes if and only if,
for each prime power $q$, there is an integer $m_q$ such that a 3-connected
member of ${\mathcal R}(q)$ has at most $m_q$ inequivalent
GF(7)-representations.
  The theorems on inequivalent representations of matroids are consequences of
structural results that do not rely on representability. The bulk of this paper
is devoted to proving such results.
