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  "body": "\n## Description\n\nA function $a \\colon \\Q \\to \\R$ has *left-finite support* if for every $q \\in \\Q$ there are only finitely many $r \u003c q$ with $a(r) \\ne 0$.\nThe Levi-Civita field $\\mathcal{R}$ is the set of such functions, written as formal sums $a = \\sum_{q \\in \\Q} a(q)\\, \\varepsilon^q$.\nAddition is termwise, and the product is $(ab)(q) = \\sum_{r + s = q} a(r) b(s)$, a finite sum because both supports are left-finite.\nFor $a \\ne 0$, let $q_0$ be the least $q$ with $a(q) \\ne 0$; then $a \u003e 0$ means $a(q_0) \u003e 0$.\nWith these operations and this order, $\\mathcal{R}$ is an [[ordered field|010JCV9J4E]] that contains $\\R$ as the constant series.\n\n## Remarks\n\nCheck that $\\varepsilon = 1 \\cdot \\varepsilon^1$ is a positive infinitesimal: $\\varepsilon \u003e 0$ since its leading coefficient is $1$, and for each integer $n \\ge 1$ the element $1/n - \\varepsilon$ has leading term $\\frac{1}{n} \\varepsilon^0$, so it is positive.\nHence $\\mathcal{R}$ is not [[Archimedean|0HQARMKEV5]], and $\\varepsilon^{-1}$ is larger than every real number.\nSince exponents are rational, $\\varepsilon$ has roots of every order, for example $(\\varepsilon^{1/2})^2 = \\varepsilon$.\nThe field $\\mathcal{R}$ is [[real closed|778KFSM09P]], and it is Cauchy complete for the topology of its order.\nAdjoining a square root of $-1$ gives an algebraically closed field.\n\n## Explanations\n\n### L1 human\n\nThe Levi-Civita field adds to the real numbers an infinitely small positive number $\\varepsilon$ and its rational powers.\nIts elements are series in $\\varepsilon$ such as $3 + 2\\varepsilon^{1/2} - \\varepsilon^{5}$, with at most finitely many terms below each power.\nA number is positive when its biggest term, the one with the lowest power of $\\varepsilon$, is positive.\nSuch fields are used to compute with infinitesimals exactly, for example in automatic differentiation.\n",
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