Forms of pointed Hopf algebras

Using descent theory, we study Hopf algebra forms of pointed Hopf algebras. It turns out that the set of isomorphism classes of such forms are in one- to-one correspondence to other known invariants, for example the set of iso morphism classes of Galois extensions with a certain group F, or the set of isometry classes of m-ary quadratic forms. Our theory leads to a classific ation of all Hopf algebras over a field of characteristic zero that become pointed after a base extension, in dimension p, p(2) and p(3), with p odd.