Let $(A,m)$ be an artinian local ring with residue field $k$.

- Suppose $k$ is algebraically closed. Is $A$ necessarily a $k$-algebra? If not what are some simple counterexamples?
- Suppose $k$ has characteristic $0$ (not necessarily algebraically closed). Is $A$ necessarily a $\mathbb{Q}$-algebra? If not what are some simple counterexamples?

EDIT: Later I realized this question is very much related to this one. I think it's still slightly simpler and might get an answer so I will not delete it yet.