  Chevalley's theorem states that every smooth connected algebraic group over a
perfect field is an extension of an abelian variety by a smooth connected
affine group. That fails when the base field is not perfect. We define a
pseudo-abelian variety over an arbitrary field k to be a smooth connected
k-group in which every smooth connected affine normal k-subgroup is trivial.
This gives a new point of view on the classification of algebraic groups: every
smooth connected group over a field is an extension of a pseudo-abelian variety
by a smooth connected affine group, in a unique way.
  We work out much of the structure of pseudo-abelian varieties. These groups
are closely related to unipotent groups in characteristic p and to
pseudo-reductive groups as studied by Tits and Conrad-Gabber-Prasad. Many
properties of abelian varieties such as the Mordell-Weil theorem extend to
pseudo-abelian varieties. Finally, we conjecture a description of
Ext^2(G_a,G_m) over any field by generators and relations, in the spirit of the
Milnor conjecture.
