Algebraic second cohomology group for trivial group action of additive group of a field on additive group of a field

Suppose $$K$$ is a field. Denote by $$G_a$$ the additive group of $$K$$. The underlying set of $$G_a$$ is the same as the underlying set of $$K$$. We wish to describe here the algebraic second cohomology group:

$$H^2_{alg}(G_a;G_a)$$

for the trivial action of $$G_a$$ on itself.

This will also allow for a classification of the two-dimensional unipotent algebraic groups over $$K$$

Description of the cohomology group
The following are true:


 * When $$K$$ has characteristic zero, this cohomology group is trivial, and thus, the only extension possible is isomorphic to the additive group of the two-dimensional vector space over $$K$$.
 * When $$K$$ has characteristic equal to a prime number $$p$$, this cohomology group is itself isomorphic to $$G_a$$.

Related facts

 * Classification of connected unipotent two-dimensional algebraic groups over an algebraically closed field

Elements
When $$K$$ has characteristic equal to a prime number $$p$$, we can use the cohomology class of the following element as a generator for the cohomology group:

$$(X,Y) \mapsto - \sum_{i=1}^{p-1} \frac{(p-1)!}{i!(p-i)!} X^iY^{p-i}$$

The other cohomology classes are obtained by multiplying this cohomology class by field elements.

Although there is a total of $$|G_a|$$ many cohomology classes, there are only two "types" of cohomology classes under the equivalence relation of being in the same orbit, and hence only two "types" of group extensions up to pseudo-congruence:

Note that this cocycle is a valid cocycle even for fields not of characteristic $$p$$, but for such fields, it is an algebraic coboundary. Explicitly, it is the coboundary of the 1-cochain $$X \mapsto (1/p)X^p$$. Thus, it gives the trivial cohomology class in these cases.