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  "body": "\n## Summary\n\nFix a prime $p$ and a [[commutative ring|0310REJJY8]] $A$.\nThe ring $W(A)$ of $p$-typical Witt vectors consists of infinite sequences of elements of $A$ with a special addition and multiplication.\nThese operations are built so that certain polynomial functions of the entries, the ghost components, turn them into ordinary addition and multiplication.\nThe construction lifts rings of characteristic $p$ to characteristic $0$: the Witt vectors of the [[finite field|0ZXZ82FTX2]] $\\F_p$ form the ring $\\Z_p$ of [[p-adic integers|V6TJAFZ4MD]].\nWitt vectors are a basic tool in arithmetic geometry and in the theory of [[perfect fields|VWBVKT794W]] of characteristic $p$.\n\n## Explanations\n\n### L1 human\n\nA $p$-adic integer can be written as a digit expansion $a_0 + a_1 p + a_2 p^2 + \\cdots$ with digits in $\\set{0, \\ldots, p-1}$.\nAdding two such expansions needs carries, and the carry rule does not come from the arithmetic of the digits modulo $p$ alone.\nWitt vectors make the carry rule into universal polynomials, so the same construction works with \"digits\" taken from any ring.\nFor example, in $W(\\F_2)$ the sum of $(1, 0, 0, \\ldots)$ with itself is $(0, 1, 0, \\ldots)$, just as $1 + 1 = 0 + 1 \\cdot 2$ in base $2$.\n",
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