I have been studying for my Qualifying Exam and came across the following problem:
Let $R\subseteq T$ be integral domains and suppose that $a\in T$ satisfies a monic polynomial of degree $d$ with coefficients in $R$. Let $S$ denote the intersection of all subrings of $T$ containing $R$ and $a$.
In part (a), I showed that $$ S=\{r_0+…+r_{d-1}a^{d-1}|r_i\in R \hspace{.1cm}\forall i\} $$
Part (b) states: Prove that if $Q$ is a maximal ideal of $R$,then here there are at most $d$ maximal ideals $P\subseteq S$ with $P\cap R=Q$.
I have worked out the following: There are at most $d$ irreducible factors of $\overline{f(x)}$ in $(R/Q)[x]$ where $\overline{f(x)}$ is $f(x)$ with coefficients reduced mod $Q$. Since $Q$ is maximal, $(R/Q)$ is a field hence, the maximal ideals of $(R/Q)[x]$ are irreducible polynomials. Thus we have at most $d$ maximal ideals of $(R/Q)[x]$ containing $<\overline{f(x)}>$. However, I am confused about how I relate that back to the maximal ideals of $S$ containing $Q$.
Is it true that $$ S=R[a]\cong R[x]/<f(x)> $$
even though $f(x)$ is not irreducible? If so, then can we look at $$ R[x]/<f(x)>/Q=(R/Q)[x]/<\overline{f(x)}> $$ and the maximal ideals of $(R/Q)[x]/<\overline{f(x)}>$ are the maximal ideals of $(R/Q)[x]$ containing $<\overline{f(x)}>$?
Any insight to Ring Theory and ideals will be helpful. Thank you in advance!