Let $S$ be a local integral domain and $S[X]$ be a polynomial ring.

Choose $f, g$ from $S[X]$ as follows:

$f:= X^n + c_{n-1}X^{n-1} + ... + c_1X + c_0$

$g:= a_mX^m + ... + a_0$,

where $a_0, a_1,...,a_m$ all lie in the unique maximal ideal $m_S$ of $S$.

If $g$ is irreducible, does the ideal $(f,g)$ contain a non-trivial element other than zero of $S$?

Let $S$ be a local integral domain and $S[X]$ be a polynomial ring.

Choose $f, g$ from $S[X]$ as follows:

$f:= X^n + c_{n-1}X^{n-1} + ... + c_1X + c_0$

$g:= a_mX^m + ... + a_0$,

where $a_0, a_1,...,a_m$ all lie in the unique maximal ideal $m_S$ of $S$.

If $g$ is irreducible, does the ideal $(f,g)$ contain a non-trivial element of $S$ other than zero?

Let S be a local integral domain and S[X] be a polynomial ring.

Choose f, g from S[X] as follows:

f:= X^n + c_{n-1}X^{n-1} + ... + c_1X + c_0

g:= a_mX^m + ... + a_0,

where a_0, a_1,...,a_m all lie in the unique maximal ideal m_S of S.

Q. If g is irreducible, does the ideal (f,g) contain a non-trivial element other than zero of S?

Let $S$ be a local integral domain and $S[X]$ be a polynomial ring.

Choose $f, g$ from $S[X]$ as follows:

$f:= X^n + c_{n-1}X^{n-1} + ... + c_1X + c_0$

$g:= a_mX^m + ... + a_0$,

where $a_0, a_1,...,a_m$ all lie in the unique maximal ideal $m_S$ of $S$.

If $g$ is irreducible, does the ideal $(f,g)$ contain a non-trivial element other than zero of $S$?