$(O_K/pO_K)^p＝O_K/pO_K$holds, then $∃b∈K^×$, such that $｜a-b^p｜≦｜p｜$

Question

Let $L$ be finite extension of $ \Bbb{Q}_p$ and field $K$ satisfies $L⊆K⊆\Bbb{C}_p$.Let $O_K$ be ring of integers of $K$. Suppose $(O_K/pO_K)^p＝O_K/pO_K$・・・① holds, then $∃b∈K^×$, such that $｜a-b^p｜≦｜p｜$. How can I take such $b$?

I want to use the condition ①(in particular surjectivity to prove existence of $b$), but $｜｜$ disturbs me. Thank you in advance.

Answer 1

$b$ is any element of $O_K$ such that $b^p=a\in O_K/(p)$.

$b^p -a=0\in O_K/(p)$ means that $b^p-a\in (p)$ ie. $|b^p-a|\le |p|$