Let $\mathbb{F}_p$ be a finite field with $p$ elements, and $\kappa := \mathbb{F}_q$ an extension of $\mathbb{F}_p$ of degree $n$ with $q = p^n$. Then $W_\infty(\kappa)$ is a ring extension of $W_\infty(\mathbb{F}_p) = \mathbb{Z}_p$, the ring of $p$-adic integers.
I believe that $W_\infty(\kappa) = \mathbb{Z}_p [ \mu_{q-1} ] $ where $ \mu_{q-1}$ is the primitive $(q-1)$-th root of unity obtained by lifting the root of $X^{q}- X $ over $\mathbb{F}_p$ using Hensel's Lemma. Denote $\mu := \mu_{q-1}$.
Please clarify a question regarding the trace map $Tr:Z_p[\mu]×Z_p[\mu] \to Z_p$. The matrix of μ is a permutation matrix in the basis $1, \mu, \mu^2, \ldots, \mu^{q-2}$ in which case the trace of $\mu^i$ is 0 for 1≤i≤q−2, and the trace of identity is $q−1$.
But the above computation seems dubious to me. Can someone clarify where I am going wrong?
Thank you.