I was working on the irreducibility of cubic equations over a non specific field (at first) and came up with this question: Given a cubic polynomial $d(t)=t^3+at^2+bt+c$ in $k[x]$ where $char(k)=0$, when does $d$ has any solutions in $k$. I know that if $k=\mathbb{F}_p$, the answer could be "never" (depending on $p$ and the coefficients of $d$). If $k=\mathbb{R}$, the answer is "always" and if $k=\mathbb{Q}$ then "sometimes".
Of course $k=\bar{k}$ does the job, but I'm looking (hopefully) for a weaker condition over $k$. I'm guessing $k$ containing square and cubic roots does the job, due to the known formulae involving the solution of a cubic. Thanks a lot!