On the Galois and flat cohomology of unipotent algebraic groups over local and global function fields. I

We discuss some results on the triviality and finiteness for Galois cohomology of connected unipotent groups over non-perfect (and especially local and global function) fields, and their relation with the closedness of orbits, which extend some well known results of Serre, Raynaud and Oesterlé. As one of the applications, we show that a separable additive polynomial over a global field k of characteristic p>0 in two variables is universal over k if and only if it is so over all completions kv of k.