  The well-known formula $det(A\cdot B)=\det A \cdot \det B$ can be easily
proved for finite dimensional matrices but it may be incorrect for the
functional determinants of differential operators, including the ones which are
relevant for Quantum Field Theory applications. Considerable work has been done
to prove that this equality can be violated, but in all previously known cases
the difference could be reduced to renormalization ambiguity. We present the
first example, where the difference between the two functional determinants is
a nonlocal expression and therefore can not be explained by the renormalization
ambiguity. Moreover, through the use of other even dimensions we explain the
origin of this difference at qualitative level.
