 Solution

Suppose $x\notin {\mathcal{O}}_K$, so $v(x)<0$. The ultrametric inequality gives $v(1+x)=v(x)$. If $1+x=y^n$, thenFor a discrete valuation, only finitely many positive integers $n$ divide the fixed nonzero integer $v(x)$. Therefore $x$ cannot belong to $S$, proving $S\subseteq {\mathcal{O}}_K$.

Solved by gpt-5.6-sol high.