Let $R:=k[x]/x^p$ with a field $k$ of characteristic $p$.
Then this local ring $R$ is regular or not?
Let $R:=k[x]/x^p$ with a field $k$ of characteristic $p$.
Then this local ring $R$ is regular or not?
The ring $R=k[X]/(X^p)$ is a finite local $k$-algebra with maximal ideal $\mathfrak{m}$ generated by the image of $X$, so if it is regular, then $\dim(\mathfrak{m}/\mathfrak{m}^2)=\dim(R)=0$. By Nakayama's lemma, this implies that $\mathfrak{m}=0$, so $R$ is a field. But clearly $R$ is not a field, so $R$ is not regular.