Let $F,G\in K[x,y]$ be two curves such that $F(p)=G(p)=0$. If $F,G$ have no common component then the set $\{F^n: n\in \mathbf{N}\}$ of polynomials is linearly independent in the $K$-vector space $O_p/\langle G\rangle$ where $O_p$ is the local ring of all rational functions defined at $p$.
My attempt went as follows:
If $\alpha_1 F^{n_1}+\ldots+\alpha_mF^{n_m}=0$ in $O_p/\langle G\rangle$ where $\alpha_i$'s are elements of $K$ then, $\alpha_1 F^{n_1}+\ldots+\alpha_mF^{n_m}=\frac{h}{k}$ for some $h,k\in K[x,y]$ such that $k(p)\not=0$. Hence, $\alpha_1 kF^{n_1}+\ldots+\alpha_m kF^{n_m}=hG$. Now, since $K[x]$ is a PID and hence a UFD so, $K[x,y]=K[x][y]$ is also a UFD, since the polynomial ring over a Unique factorization domain is a unique factorization domain. Let $m=\operatorname{min}_{i=1}\{n_i\}$. Let $R$ be any prime factor of $F^m$ then, $R$ is also a prime factor of $hG$. Now, if all prime factors of $F$ do not divide $G$ then they divide $h$ and thus, $h=F^mH$ for some $H\in K[x,y]$.
but I can not complete it. Any help? And by the way, how to deduce that if $F$ and $G$ have common component then $dim_K O_p/<F,G>$ is infinite?