This problem has two parts, the first I did it. But I have problems with the second.
Given an irreducible polynomial F over $k[T]$. Prove that the ring $ k\left[ {T_1 ,...,T_n } \right]/\left( {I\left( {V\left( F \right)} \right)} \right) $, is an integral domain. Where given a variety W, we denote $I(W)$ the ideal of all the polynomials over $k[T_1,...,T_n]$ that vanish in W, and $ V\left( F \right) = \left\{ {\left( {x_1 ,...,x_n } \right) = x \in K^n ;F\left( x \right) = 0} \right\} $ , where K it´s probably algebraically closed.
But now... The second part :
Since it's an integral domain, it field of fractions exist. How can we prove the following:
Let $F(T_1,...,T_n) = G(T_1,...,T_n)+H(T_1,...,T_n)$ where G is a homogeneous polynomial of degree $m-1$ and H is a homogeneuous polynomial of degree $m$. Assume that F is irreducible in $k[T_1,...,T_n]$ where k is a field.
$ Frac\left( {k\left[ {T_1 ,...,T_n } \right]/\left( {I\left( {V\left( F \right)} \right)} \right)} \right) $ it´s isomorphic to a field of rational functions $ k\left( {T_1 ,...,T_m } \right) $ for some m.
I have no idea how to use the hypothesis of being sum of two irreducible homogeneous polynomials.