  We say that a set $S$ is additively decomposed into two sets $A$ and $B$, if
$S = \{a+b : a\in A, \ b \in B\}$. Here we study additively decompositions of
multiplicative subgroups of finite fields. In particular, we give some
improvements and generalisations of results of C. Dartyge and A. Sarkozy on
additive decompositions of quadratic residues and primitive roots modulo $p$.
We use some new tools such the Karatsuba bound of double character sums and
some results from additive combinatorics.
