I am working through Eisenbud's Commutative Algebra and am stuck on a "trivial" part of an exercise.
Let $G$ be a finite abelian group that acts by characters on $S = k[x_1,\ldots,x_r]$, where $k$ is a field. I was asked to prove that the ring $S^G$ of invariants of $G$ are generated by monomials $\prod_{i=1}^r x_i^{a_i}$ whose exponent vectors are in the kernel of a map from ${\bf Z}^r$ to a finite abelian group. I've done this part.
Now it says to conclude that the quotient field of $S^G$ is isomorphic to a field of rational functions in $r$ variables. I looked at the answer key and it simply says that the field of rational functions will be $k(t_1,\ldots,t_r)$ where $t_1,\ldots,t_r$ are somewhat mysteriously constructed, at least to me. The way it is phrased, the reasoning should be somewhat obvious, but I am not seeing it straightaway and I would appreciate if someone could spell it out for me. Thanks!