I am wondering if we can find an irreducible polynomial $g(x)$ in $\mathbb{Z}[x]$ such that
- The constant term, $c(g)=\pm 1$ and the leading coefficient $\ell(g)=\pm 1$,
- the ideal generated by $g(x)$ and $5x+7$ is the ring $\mathbb{Z}[x]$, that is, $(g(x),5x+7)=1$, in other words: $g(-7/5)=\pm 1$,
- the ideal generated by $g(x)$ and $2x-3$ is the ring $\mathbb{Z}[x]$, that is, $(g(x), 2x-3)=1$, in other words: $g(3/2)=\pm 1$.
Thanks.
PS: You can change $5x+7$ and $2x-3$ with any polynomials such that their constant terms and the leading coefficients are not units in $\mathbb{Z}$.