Witt vectors

Remarks

The first sum polynomials are $S_0 = X_0 + Y_0$ and $S_1 = X_1 + Y_1 + \frac{X_0^p + Y_0^p - (X_0 + Y_0)^p}{p}$, which has integer coefficients. The ghost map $a \mapsto (w_0(a), w_1(a), \ldots)$ is a ring homomorphism $W(A) \to A^{\N}$, and it is an isomorphism when $p$ is a unit in $A$. A ring homomorphism $f \colon A \to B$ induces $W(f) \colon W(A) \to W(B)$ by applying $f$ to each entry.

Example: $W(\F_p)$ is isomorphic to the ring $\Z_p$ of p-adic integers. For $p = 2$, compute $(1, 0, \ldots) + (1, 0, \ldots)$ in $W(\Z)$ through ghost components: $w_0 = 1 + 1 = 2$ and $w_1 = 1 + 1 = 2$. So $a_0 = 2$ and $a_0^2 + 2 a_1 = 2$, which gives $a_1 = -1$. Reducing modulo $2$, in $W(\F_2)$ the sum is $(0, 1, \ldots)$, which corresponds to $2 \in \Z_2$.

Non-example: the componentwise addition on $\F_2^{\N}$ gives $(1, 0, \ldots) + (1, 0, \ldots) = (0, 0, \ldots)$. This is not the Witt sum, because it does not respect the ghost components $w_1$ over $\Z$.

The Teichmüller representative of $x \in A$ is $[x] = (x, 0, 0, \ldots)$, and it satisfies $[xy] = [x][y]$.