I am reading Fulton's book on algebraic curves.In section $2.1$ viz. coordinate-ring ,there is a problem which is given below:
Assume $k$ is algebraically closed.Let $F\in k[X,Y]$ be irreducible and monic in $Y$ i.e. $F=Y^n+a_1(X)Y^{n-1}+\dots+a_n(X)$ and $n>0$.Suppose $V=Z(F)\subset \mathbb A^2(k)$.Then the natural map $\varphi:k[X]\to k[X,Y]/\langle F\rangle=\Gamma(V)$ is one-one(so that $k[X]$ can be regarded as a subring of $\Gamma(V)$ ) and show that $\overline{1},\overline{Y},...,\overline{Y}^{n-1}$ generate $\Gamma(V)$ as a $k[X]$-module.
I need some hint in order to solve this problem.I am primarily stuck with proving that $Ker(\varphi)=\{0\}$ because $f(X)\in Ker(\varphi)$ means that $f(X)\in \langle F\rangle$ but form there how can I show that $f(X)=0$.
If $f(X)\in \langle F\rangle$ i.e. if $f(X)=FG$ for some $G\in k[X,Y],$ prove by contradiction that $G=0.$ For this, reason on the $Y$-degree of $f,F,G.$