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  "body": "\n## Definition\n\nLet $p$ be a prime number and let $A$ be a [[commutative ring|0310REJJY8]].\nFor $n \\ge 0$, the $n$-th ghost polynomial is\n$$w_n(X_0, \\ldots, X_n) = \\sum_{i=0}^{n} p^i X_i^{p^{n-i}} = X_0^{p^n} + p X_1^{p^{n-1}} + \\cdots + p^n X_n.$$\nThere are unique sequences of [[polynomials|0SDC934R52]] $S_n, P_n \\in \\Z[X_0, \\ldots, X_n, Y_0, \\ldots, Y_n]$ such that for every $n \\ge 0$:\n$$w_n(S_0, \\ldots, S_n) = w_n(X) + w_n(Y), \\qquad w_n(P_0, \\ldots, P_n) = w_n(X) \\, w_n(Y).$$\nThe ring of *$p$-typical Witt vectors* $W(A)$ is the set $A^{\\N}$ of sequences $a = (a_0, a_1, a_2, \\ldots)$ with entries in $A$, with the operations\n$$(a + b)_n = S_n(a_0, \\ldots, a_n, b_0, \\ldots, b_n), \\qquad (ab)_n = P_n(a_0, \\ldots, a_n, b_0, \\ldots, b_n).$$\nWith these operations $W(A)$ is a commutative ring with zero $(0, 0, \\ldots)$ and identity $(1, 0, 0, \\ldots)$.\n\n## Remarks\n\nThe 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.\nThe ghost map $a \\mapsto (w_0(a), w_1(a), \\ldots)$ is a [[ring homomorphism|0VTH8KPCG2]] $W(A) \\to A^{\\N}$, and it is an isomorphism when $p$ is a unit in $A$.\nA ring homomorphism $f \\colon A \\to B$ induces $W(f) \\colon W(A) \\to W(B)$ by applying $f$ to each entry.\n\nExample: $W(\\F_p)$ is isomorphic to the ring $\\Z_p$ of [[p-adic integers|V6TJAFZ4MD]].\nFor $p = 2$, compute $(1, 0, \\ldots) + (1, 0, \\ldots)$ in $W(\\Z)$ through ghost components: $w_0 = 1 + 1 = 2$ and $w_1 = 1 + 1 = 2$.\nSo $a_0 = 2$ and $a_0^2 + 2 a_1 = 2$, which gives $a_1 = -1$.\nReducing modulo $2$, in $W(\\F_2)$ the sum is $(0, 1, \\ldots)$, which corresponds to $2 \\in \\Z_2$.\n\nNon-example: the componentwise addition on $\\F_2^{\\N}$ gives $(1, 0, \\ldots) + (1, 0, \\ldots) = (0, 0, \\ldots)$.\nThis is not the Witt sum, because it does not respect the ghost components $w_1$ over $\\Z$.\n\nThe Teichmüller representative of $x \\in A$ is $[x] = (x, 0, 0, \\ldots)$, and it satisfies $[xy] = [x][y]$.\n",
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