This may be helpful.
In the below paper it is proved that every element in the full matrix algebra $M_3(k)$ is a sum of two idempotents if and only if every polynomial of degree $3$ has a root in $k$.
de Seguins Pazzis, Clément; On decomposing any matrix as a linear combination of three idempotents. Linear Algebra Appl. 433 (2010), no. 4, 843–855.
Added: the name "cubically closed field" was almost suggested by a removed comment of @EricWofsey. And it seems that it is the right name for these fields.