Let $R$ be a Dedekind ring and consider $R[x]$. Let $G$ be a finite group acting on $R[x]$ which acts on $R$ trivially. I want to know if there is a general method to calculate $R[x]^G$.
I unfortunately have no background in the invariant theory, and it looks like almost all notes on the invariant theory that I find on Google discuss the symmetric group $S_n$ acting on $k[x_1, ..., x_n]$ the polynomial ring of $n$ variables over a field $k$. I'm not sure if my question would be answered in much later chapters of a typical invariant theory textbook. Even if I strengthen my polynomial ring to be $k[x]$, I'm concerned with a finite group action other than $S_n = S_1$.
Could someone guide me on where to start learning about theories discussing my question?
Thank you.