I'm reading the construction of the Witt ring from Lang's algebra. This is a series of exercises in chap. VI. In exercise 50 he says:
If $x\in W_n(k)$ show that there exists $\xi\in W_n(\bar{k})$ such that $\mathfrak{p}(\xi)=x$. Do this inductively solving first for the first component and then showing that a vector $(0,a_1,...,a_{n-1})$ is in the image of $\mathfrak{p}$ if and only if $(a_1,...,a_{n-1})$ is in the image of $\mathfrak{p}.$
$\mathfrak{p}$ denotes the Artin-Schreier operator on $W_n(A)$: $\mathfrak{p}(a)=a^p-a$. I don't see how his hint gives the solution for a general $x$. I see that it gives the solution only for $x=(0,...,0,x_0)$.