2-truncated Witt vectors over $\Bbb F_2((t))$

Let $p \in \Bbb Z$ be prime and $R$ a commutative ring. For $x = (x_0, x_1, \ldots) \in R^{\Bbb N}$, denote $x^{(n)} = \sum\limits_{i = 0}^n x_i^{p^{n-i}} p^i$. It is known that for $S = \Bbb Z[x_0, x_1, \ldots, y_0, y_1, \ldots]$ there exist sequences of polynomials $\alpha, \pi \in S^{\Bbb N}$ such that $\alpha^{(n)} = x^{(n)} + y^{(n)}$ and $\pi^{(n)} = x^{(n)} y^{(n)}$ for $x = (x_i)$, $y = (y_i)$ and every $n \ge 0$. In particular: $\alpha_0 = x_0 + y_0$, $\pi_0 = x_0 y_0$; $\alpha_1 = x_1 + y_1 - \sum\limits_{i = 1}^{p-1} {p\choose i} x_0^i y_0^{p-i}$, $\pi_1 = x_0^p y_1 + y_0^p x_1 + p x_1 y_1$.

The $(n+1)$-truncated ring $W$ is the Cartesian product $R^{n+1}$, endowed with addition $a+b = (\alpha_0(a,b), \alpha_1(a,b), \ldots, \alpha_n(a,b))$ and multiplication $ab = (\pi_0(a,b), \pi_1(a,b), \ldots, \pi_n(a,b))$, its unity is $(1,0,\ldots,0)$. For concreteness, we take $n = 1$, $p = 2$ and $R = F =$ the algebraic closure of $\Bbb F_2((t))$.

Notes: if one instead takes $R = \Bbb F_p$ and does not truncate the construction (sets $n = \infty$), then $W \cong \Bbb Z_p$, the ring of $p$-adic integers.