Let k be a field, X0 an object
(e.g., scheme, group scheme) defined over k. An object X of the same type and
isomorphic to X0 over some field K ⊃ k is called a form of X0. If k is not perfect,
both the affine line A1 and its additive group Ga have nontrivial sets of
forms, and these are investigated here. Equivalently, one is interested in
k-algebras R such that K ⊗kR≅K[t] (the polynomial ring in one variable)
for some field K ⊃ k, where, in the case of forms of Ga,R has a group (or
co-algebra) structure s : R → R ⊗kR such that (K ⊗ s)(t) = i ⊗ 1 + 1 ⊗ t. A
complete classification of forms of Ga and their principal homogeneous spaces is
given and the behaviour of the set of forms under base field extension is
studied.
