  Let $p\equiv1\pmod3$ be a prime . We study several topics on additive
decompositions concerning the set $C_p$ of all non-zero cubes in the finite
field of $p$ elements. For example, we show that when $p>184291$ , the set
$C_p$ has no decomposition of the form $C_p=A+B+C$ with $|A|,|B|,|C|\ge2$.
