  We classify one-dimensional symmetry-protected topological (SPT) phases
protected by dipole symmetries. A dipole symmetry comprises two sets of
symmetry generators: charge and dipole operators, which together form a
non-trivial algebra with translations. Using matrix product states (MPS), we
show that for a $G$ dipole symmetry with $G$ a finite abelian group, the
one-dimensional dipolar SPTs are classified by the group $H^2[G\times
G,U(1)]/H^2[G,U(1)]^2$. Because of the symmetry algebra, the MPS tensors
exhibit an unusual property, prohibiting the fractionalization of charge
operators at the edges. For each phase in the classification, we explicitly
construct a stabilizer Hamiltonian to realize the SPT phase and derive the
response field theories by coupling the dipole symmetry to background tensor
gauge fields. These field theories generalize the Dijkgraaf-Witten theories to
twisted finite tensor gauge theories.
