Let $Q_2$ \mathbb{Q}_2$ be the field of 2-adic $2$-adic numbers and $Z_2$ \mathbb{Z}_2$ its integer ring. Let $O$ be the unramified extension of $Z_2$, of degree $d$. Suppose $d\geq 3$. Denote $W$ be the roots of unity in $O$. Assign $\alpha,\beta$ to be two units of $O$, where $\beta\notin $Z_2$. \mathbb{Z}_2$. Denote $\alpha W={x\in $\alpha W=\{x\in O: x=\alpha w,w\in W }$, \},$$ $Tr$ as the trace function from $O$ to $Z_2$.\mathbb{Z}_2$.
If the degree $d$ is large enough, can we conclude that ${Tr(x)\mod $\{Tr(x)\mod 4: Tr(\beta x)\equiv0\mod2, x)\equiv 0 \mod2, x\in \alpha W }={0,1,2,3}? $\}=\{0,1,2,3\}?$$
Generally, is there any general result about distribution of trace of roots of unity in unramified extensions of a $p$-adic ring?
