Is there a field $k$ in which $x^3+3$ has double roots?

Question

I am doing a problem in which I am looking at the polynomial $x^3+3$ in a field $k$. I have separated my answer in three cases:

1. $x^3+3$ has no roots in $k$
2. $x^3+3$ has exactly one root in $k$
3. $x^3+3$ has three roots in $k$

For the last case, however, I came across a question. Is it possible that $x^3+3$ has a double (or even triple) root in a field $k$?. If I take "easy" fields such as $\mathbb{C}$ then I know it's not the case, but perhaps in some more complicated ones it happens. I have been thinking about finite fields for example but didn't find any in which this happens. Can someone help me find me a field $k$ of this form, or show me that it cannot exist?

Answer 1

The discriminant of $x^3+3$ is $\Delta=-27\cdot 3^2$. If there is a repeated root, then $\Delta=0$, so in this field $3^5=0$ and hence $3=0$. The converse is clearly true, so the result is any field $k$ with characteristic $3$.