  A question of Bergman asks whether the adjoint of the generic square matrix
over a field can be factored nontrivially as a product of square matrices. We
show that such factorizations indeed exist over any coefficient ring when the
matrix has even size. Establishing a correspondence between such factorizations
and extensions of maximal Cohen--Macaulay modules over the generic determinant,
we exhibit all factorizations where one of the factors has determinant equal to
the generic determinant. The classification shows not only that the
Cohen--Macaulay representation theory of the generic determinant is wild in the
tame-wild dichotomy, but that it is quite wild: even in rank two, the
isomorphism classes cannot be parametrized by a finite-dimensional variety over
the coefficients. We further relate the factorization problem to the
multiplicative structure of the $\Ext$--algebra of the two nontrivial rank-one
maximal Cohen--Macaulay modules and determine it completely.
