A principal homogeneous space (also called a torsor) for a group 
is a nonempty set 
equipped with a group action
of 
that is both free and transitive.
Equivalently, for every 
, there is a unique 
such that 
. After a point 
is chosen, the map 
is a bijection
from 
to 
.
Without this choice, however, 
has no distinguished point corresponding to the identity
element of 
.
For an algebraic group 
over a field 
, an algebraic principal homogeneous space is a variety 
with a compatible group action such that 
, given by 
, is an isomorphism
and 
acquires a point after a faithfully flat extension of 
. A 
-rational point makes the
space trivial by identifying it with 
. The isomorphism classes of principal homogeneous spaces under
are classified by the first Galois cohomology
set 
(Skorobogatov 2001).
Principal homogeneous spaces under an elliptic curve 
that have points over every completion of 
occur in the Tate-Shafarevich
group; a nontrivial such space has no 
-rational point.
