  We study the classical and quantum cosmology of a 4+1-dimensional space-time
with a non-zero cosmological constant coupled to a self interacting massive
spinor field. We consider a spatially flat Robertson-Walker universe with the
usual scale factor $R(t)$ and an internal scale factor $a(t)$ associated with
the extra dimension. For a free spinor field the resulting equations admit
exact solutions, whereas for a self interacting spinor field one should resort
to a numerical method for exhibiting their behavior. These solutions give rise
to a degenerate metric and exhibit signature transition from a Euclidean to a
Lorentzian domain. Such transitions suggest a compactification mechanism for
the internal and external scale factors such that $a\sim R^{-1}$ in the
Lorentzian region. The corresponding quantum cosmology and the ensuing
Wheeler-DeWitt equation have exact solutions in the mini-superspace when the
spinor field is free, leading to wavepackets undergoing signature change. The
question of stabilization of the extra dimension is also discussed.
