I have the following question:
Let $V$ be an irreducible affine variety over an algebraically closed field $K$. For any point $p \in V$, the ring $P(V) := ${$f \in K(V) | f$ is regular at $p$} is local. Denote by $M_p$ the maximal ideal of $P(V)$. Let $V = V (y − x^2) \subseteq K^2$ and $p = (0, 0)$. Show that $\dim_K (M_p/M_p^2)$ is $1$.
I have come across a way this can be solved using a geometric interpretation where $M_p/M_p^2$ is the dimension of the tangent space to $V$ at $p$. However I think that is outside the scope of my course, and I find it difficult to prove that way.
I believe I am missing something simple here. If someone could point me in the right direction it would be appreciated.