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? 1 # Trace of roots of unity in the ring of p-adic integers Let$Q_2$be the field of 2-adic numbers and$Z_2$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 $Z_2$. Denote $\alpha W={x\in O: x=\alpha w,w\in W }$, $Tr$ as the trace function from $O$ to $Z_2$.
If the degree $d$ is large enough, can we conclude that ${Tr(x)\mod 4: Tr(\beta x)\equiv0\mod2, x\in \alpha W }={0,1,2,3}?$
Generally, is there any general result about distribution of trace of roots of unity in unramified extensions of a $p$-adic ring?