Let $L$ be finite extension of $\Bbb{Q}_p$ and $K$ be a finite field. $W(K)_L$ be ramified witt ring of $K$.
I want to prove $W(K)_L$ has characteristic $0$.
Here, we admit $W(K)_L$ is integral domain.
Suppose $l$ is prime except for $p$, then $l(1,0,0,・・・)＝(l,・・・)$ is unit in $W(K)_L$, so it's never zero.
The last thing I should check is $p(1,0,0,・・・)$ is never zero.
Let $L/\Bbb{Q}_p$ be extension of ramified index $e$.
Then, $p(1,0,0,・・・)＝π^eu(1,0,0,・・・)＝u(0,0,・・・,0,1,0,・・・)$ ($u$ is invertible element of ring of integers of $L$).
Why u(0,0,・・・,0,1,0,・・・) is nonzero?
Referrence: This kind of argument and definition of ramified Witt vector is in Peter Schneider's 'Galois representation and (φ,Γ)modules'.