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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]].\nA *discrete valuation* on $K$ is a [[valuation|B85T5VKR3M]] $v \\colon K \\to \\Z \\cup \\set{\\infty}$ such that $v(K^\\times) = \\Z$.\nAn element $\\pi \\in K$ with $v(\\pi) = 1$ is a *uniformizer* of $v$.\n\n## Remarks\n\nExample: for a prime $p$, the [[p-adic valuation|04VCPS30ZE]] $v_p$ on $\\Q$ is a discrete valuation.\nIt is a valuation, and $v_p(p^n) = n$ for every $n \\in \\Z$, so it is onto $\\Z$; the prime $p$ is a uniformizer.\nExample: for a field $k$, the order of vanishing at $0$ on $k(t)$, given by $v(t^n f/g) = n$ for polynomials $f, g$ with $f(0) g(0) \\ne 0$, is a discrete valuation with uniformizer $t$.\nNon-example: the trivial valuation on a field, with $v(x) = 0$ for all $x \\ne 0$, takes values in $\\Z$ but has $v(K^\\times) = \\set{0} \\ne \\Z$.\nSome texts call a valuation discrete when its value group is isomorphic to $\\Z$; dividing by a positive generator then gives a valuation onto $\\Z$, called normalized.\nEvery $x \\in K^\\times$ is $x = u \\pi^{v(x)}$ with $v(u) = 0$.\nThe set $\\set{x : v(x) \\ge 0}$ is a [[discrete valuation ring|MGM3CNJW9V]] with maximal ideal $\\pi \\cdot \\set{x : v(x) \\ge 0}$.\n",
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