Suppose that $R_1\subseteq R_2$, and both are integral domains.
Further suppose that $R_2$ is a field, where each element $r\in R_2$ is a zero of a polynomial in $R_1[x]$ with the leading coefficient 1.
I want to show that if $a\in R_1\cap R_2$, then any polynomial in $R_1[x]$ with leading coefficient 1 and root $a^{-1}$ is reducible over $R_1$. How should I approach this problem?
Thanks in advance.
It is easy to see that $R_1$ is actually a field. Hence, $a^{-1} \in R_1$ and $x-a^{-1}$ will be a factor. Notice that this only proves reducibility if the degree is $>1$. Otherwise the claim is also false, since $x-a^{-1} \in R_1[x]$ is irreducible.