  The image of the principal minor map for n x n-matrices is shown to be
closed. In the 19th century, Nansen and Muir studied the implicitization
problem of finding all relations among principal minors when n=4. We complete
their partial results by constructing explicit polynomials of degree 12 that
scheme-theoretically define this affine variety and also its projective closure
in $\PP^{15}$. The latter is the main component in the singular locus of the 2
x 2 x 2 x 2-hyperdeterminant.
