Let be $C \subset \mathbb{A}^2 _k$ a smooth curve over algebraically closed field $k$ defined by a polynomial $f \in k[x,y]$, so $C= V(f)$.
A point $c \in C$ is called smooth if $\partial_x f$ and $\partial_y f$ aren't both zero.
Futhermore, $c \in C$ is called a regular point, if $dim \mathcal{O}_{C,c} = dim_{k(c)} m_c / m_c^2$ holds.
I heard that if $k$ is algebraically closed for a point smoothness and beeing regular is equivalent. Why? Does anybody has a good source for the proof?