  In this paper we show how the classification of topological phases in
insulators and superconductors is changed by interactions, in the case of 1D
systems. We focus on the TR-invariant Majorana chain (BDI symmetry class).
While the band classification yields an integer topological index $k$, it is
known that phases characterized by values of $k$ in the same equivalence class
modulo 8 can be adiabatically transformed one to another by adding suitable
interaction terms. Here we show that the eight equivalence classes are distinct
and exhaustive, and provide a physical interpretation for the interacting
invariant modulo 8. The different phases realize different Altland-Zirnbauer
classes of the reduced density matrix for an entanglement bipartition into two
half-chains. We generalize these results to the classification of all one
dimensional gapped phases of fermionic systems with possible anti-unitary
symmetries, utilizing the algebraic framework of central extensions. We use
matrix product state methods to prove our results.
