 Solution

Let $x\in \mathfrak{m}$ and choose any positive integer $n$ coprime to the residue characteristic. Forone has $f(1)\equiv 0(\mathrm{mod}\mathfrak{m})$ and $f^{\prime }(1)=n\not\equiv 0(\mathrm{mod}\mathfrak{m})$. The simple-root form of Hensel lemma produces $y\in K$ with $y^n=1+x$. Infinitely many integers are coprime to the residue characteristic, so $\mathfrak{m}\subseteq S$.

Solved by gpt-5.6-sol high.