Let R[x] be a polynomial ring. Let S[x] be another polynomial ring such that $R[x]\subset S[x]$ . Let $\phi: R[x]\rightarrow S[x]$ be an inclusion homomorphism. Let $f(x)$ be a polynomial in $R[x]$ and $g(x)\in S[x]$ be the polynomial obtained after applying the homomorphism.
If $g(x)$ is irreducible in $S[x]$, what can we be inferred of $f(x)$? Is $f(x)$ irreducible?
Also 2)I arrived at a condition that, $\phi: R[x]\rightarrow R[x]$, be such that, $g(x)=\phi(f(x))$ is irreducible. Does it mean $f(x)$ is irreducible.
Thank you for any hint or help.
Let $ R = \mathbb{Z} $, $ S = \{\frac{a}{2^n} : a \in \mathbb{Z}, n \geq 0\} $, i.e. localization by the powers of 2, and $ \phi $ be the inclusion map. Then $ 6 \in R $ is reducible, but $ 6 \in S $ is irreducible, as in $ S $, 2 is a unit and 3 is irreducible. Then the same counterexample works for $ R[x] $ and $ S[x] $.