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  "body": "\n## Statement\n\nLet $R$ be a [[Noetherian|0670T59HGE]] [[local|MZ02BKPHJA]] [[integral domain|0QVD7WBFGN]] of [[Krull dimension|MVGBS3J1P7]] $1$, with maximal ideal $\\mathfrak{m}$ and [[residue field|NJ3APQ8BHH]] $k = R/\\mathfrak{m}$.\nThe following conditions are equivalent:\n\n1. $R$ is a [[discrete valuation ring|MGM3CNJW9V]].\n2. $R$ is [[integrally closed|H20P600HQS]] in its field of fractions.\n3. $\\mathfrak{m}$ is a [[principal ideal|0SVYSP8HCG]].\n4. $\\dim_k \\mathfrak{m}/\\mathfrak{m}^2 = 1$.\n5. Every nonzero ideal of $R$ is a power $\\mathfrak{m}^n$ with $n \\ge 0$.\n6. There is $x \\in R$ such that every nonzero ideal of $R$ is $(x^n)$ for some $n \\ge 0$.\n\n## Remarks\n\nThe key step is that 2 implies 3.\nTake $a \\in \\mathfrak{m}$ nonzero; since $\\mathfrak{m}$ is the only nonzero prime, some power $\\mathfrak{m}^n$ lies in $(a)$; with $n$ minimal one can pick $b \\in \\mathfrak{m}^{n-1}$ with $b \\notin (a)$.\nFor $y = a/b$, the element $y^{-1}$ is not in $R$, so by integral closedness it is not integral, and then $y^{-1}\\mathfrak{m} \\not\\subseteq \\mathfrak{m}$; this forces $y^{-1}\\mathfrak{m} = R$, so $\\mathfrak{m} = (y)$.\nThe other implications use Nakayama's lemma and the fact that $\\bigcap_n \\mathfrak{m}^n = 0$.\n\nCondition 4 says that $R$ is a [[regular local ring|63XXTF6B5M]] of dimension $1$.\nExample: the local ring of $k[x, y]/(y - x^2)$ at the origin is a discrete valuation ring, with $\\mathfrak{m} = (x)$, since $y = x^2$ there.\nThe local ring of $k[x, y]/(y^2 - x^3)$ at the origin is not: its maximal ideal $(x, y)$ needs two generators, and $t = y/x$ satisfies $t^2 = x$, so it is not integrally closed.\n\nThe hypotheses matter.\nWithout dimension $1$, the ring $k[\\![x, y]\\!]$ is a Noetherian integrally closed local domain, but not a discrete valuation ring.\nWithout the Noetherian condition, a valuation ring with value group $\\Q$ is an integrally closed local domain of dimension $1$, but its maximal ideal is not principal.\n\n## Explanations\n\n### L1 human\n\nDiscrete valuation rings can be recognised in several ways.\nAmong the \"one-dimensional\" local rings, they are exactly the ones without singularity.\n\"Without singularity\" can mean that no fraction is missing which should be there.\nIt can also mean that the maximal ideal has a single generator, or that all ideals are powers of one ideal.\nThe theorem says that all these meanings agree.\n\n### L2 human\n\nThis result is the local step in the theory of [[Dedekind domains|7NPY2CN5KB]].\nA Noetherian domain of dimension $1$ is a Dedekind domain exactly when its localizations at nonzero primes are discrete valuation rings.\nIn geometry, it says that a point on a curve is nonsingular exactly when the local ring there is integrally closed, which is why normalization resolves the singularities of curves.\nCondition 4 links the theory with [[regular local rings|63XXTF6B5M]], which generalise discrete valuation rings to higher dimension.\n",
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