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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 ...

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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.

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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.

As a consequence, in a real-closed field, every polynomial splits into linear and quadratic factors. Of course, "cubically closed fields" are more general than real-closed fields, but nevertheless this might give a direction to search for characterisations, and there is a large literature on real-closed fields.

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