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  "body": "\n## Statement\n\nLet $R$ be a [[Dedekind domain|7NPY2CN5KB]], that is, a [[Noetherian|0670T59HGE]] [[integrally closed domain|H20P600HQS]] in which every nonzero [[prime ideal|00MHHZ9SNE]] is [[maximal|0PV74TR5FP]].\nLet $I$ be a nonzero [[ideal|096Z9MP2ME]] of $R$ with $I \\ne R$.\nThen there are distinct nonzero prime ideals $P_1, \\ldots, P_r$ and integers $e_1, \\ldots, e_r \\ge 1$ with\n$$I = P_1^{e_1} P_2^{e_2} \\cdots P_r^{e_r}.$$\nThe primes $P_i$ and exponents $e_i$ are unique up to the order of the factors.\n\n## Remarks\n\nOne proof shows first that every nonzero ideal contains a product of nonzero primes, using the Noetherian condition.\nThen it shows that every nonzero prime $P$ is [[invertible|E190TAV2S6]]: there is a fractional ideal $P^{-1}$ with $P P^{-1} = R$; here integral closedness and dimension $1$ are used.\nDividing $I$ by a prime that contains it gives a strictly larger ideal, and the Noetherian condition ends the process.\nUniqueness follows by cancelling invertible primes, since $P \\supseteq P_1 \\cdots P_r$ forces $P = P_i$ for some $i$.\n\nExample: in $R = \\Z[\\sqrt{-5}]$, the element $6$ factors in two ways as $2 \\cdot 3 = (1 + \\sqrt{-5})(1 - \\sqrt{-5})$, so $R$ is not a [[UFD|0W4X9G6263]].\nWith $P = (2, 1 + \\sqrt{-5})$, $Q = (3, 1 + \\sqrt{-5})$ and $Q' = (3, 1 - \\sqrt{-5})$, one has $P^2 = (2)$ and $Q Q' = (3)$, so $(6) = P^2 Q Q'$, and the two factorizations of $6$ come from regrouping these prime ideals.\n\nConversely, an integral domain in which every nonzero proper ideal is a product of prime ideals is a Dedekind domain.\nThe dimension hypothesis is needed: $k[x, y]$ over a field $k$ is Noetherian and integrally closed, but it has dimension $2$.\nIn it, the only prime ideal that contains $(x, y^2)$ is $(x, y)$, and no power of $(x, y)$ equals $(x, y^2)$, so $(x, y^2)$ is not a product of prime ideals.\n\n## Explanations\n\n### L1 human\n\nIn many rings of algebraic numbers, numbers do not factor uniquely into primes.\nDedekind's idea was to factor ideals instead of numbers.\nIn a Dedekind domain, every ideal splits into prime ideals in exactly one way, just as every integer splits into prime numbers.\nThis rescues unique factorization for all rings of integers of number fields.\n\n### L2 human\n\nThe theorem shows that the nonzero fractional ideals of a Dedekind domain form a free abelian group with the nonzero prime ideals as a basis.\nThe quotient by the principal fractional ideals is the [[ideal class group|WQ46BD3ARY]], which measures how far $R$ is from being a [[principal ideal domain|0CMD0BCH8T]].\nLocally, the exponent $e_i$ is the valuation of $I$ in the [[discrete valuation ring|MGM3CNJW9V]] $R_{P_i}$.\nIn number theory, the factorization of $pR$ for a prime number $p$ describes how $p$ splits in the number field.\n",
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