# “Degree 3 fields”

I was wondering what was known about fields $k$ having the property that any polynomial over $k$ of degree $3$ has at least one root in $k$. Does such a field have a special name ? Is there some kind of classification ?

Many thanks ...

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 $k$.
Such fields are related to real-closed fields, which satisfy an even stronger property. A field $K$ is called real-closed, if it is formally real, i.e., $-1$ is not a sum of squares in $K$, and no proper algebraic extension is formally real. Then we have the following result:
Theorem: In a real-closed field, every polynomial of odd degree $>1$ has a root.