  The prism graph is the dual of the complete graph on five vertices with an
edge deleted, $K_5\backslash e$. In this paper we determine the class of binary
matroids with no prism minor. The motivation for this problem is the 1963
result by Dirac where he identified the simple 3-connected graphs with no minor
isomorphic to the prism graph. We prove that besides Dirac's infinite families
of graphs and four infinite families of non-regular matroids determined by
Oxley, there are only three possibilities for a matroid in this class: it is
isomorphic to the dual of the generalized parallel connection of $F_7$ with
itself across a triangle with an element of the triangle deleted; it's rank is
bounded by 5; or it admits a non-minimal exact 3-separation induced by the
3-separation in $P_9$. Since the prism graph has rank 5, the class has to
contain the binary projective geometries of rank 3 and 4, $F_7$ and $PG(3, 2)$,
respectively. We show that there is just one rank 5 extremal matroid in the
class. It has 17 elements and is an extension of $R_{10}$, the unique splitter
for regular matroids. As a corollary, we obtain Dillon, Mayhew, and Royle's
result identifying the binary internally 4-connected matroids with no prism
minor [5].
