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  "body": "\n## Definition\n\nLet $a \\in \\C^n$ and let $A_a$ be a [[germ of an analytic set|A2WHCVP6JN]] at $a$.\nThe germ $A_a$ is *irreducible* if:\n\n1. $A_a$ is not the empty germ, and\n2. whenever $A_a = B_a \\cup C_a$ for germs of analytic sets $B_a, C_a$ at $a$, one has $A_a = B_a$ or $A_a = C_a$.\n\n## Remarks\n\nEquivalently, the ideal $\\mathcal{I}(A_a)$ of [[holomorphic germs|2TGPZ8VW2D]] vanishing on $A_a$ is a [[prime ideal|00MHHZ9SNE]] of $\\mathcal{O}_{n,a}$.\nExample: the germ at $0$ of the cusp $A = \\set{z_1^2 = z_2^3}$ in $\\C^2$ is irreducible.\nThe germ $f = z_1^2 - z_2^3$ is irreducible in $\\mathcal{O}_{2,0}$, since it is a Weierstrass polynomial of degree $2$ in $z_1$.\nA factorization would give a germ $h$ with $h(z_2)^2 = z_2^3$, which is impossible by comparing orders of vanishing.\nSince $\\mathcal{O}_{2,0}$ is a unique factorization domain, the ideal $(f)$ is prime, hence radical, and the [[Rückert Nullstellensatz|3VRZ80X45P]] gives $\\mathcal{I}(A_0) = (f)$, a prime ideal.\nNon-example: the germ at $0$ of $\\set{z_2^2 = z_1^2(1 + z_1)}$ is reducible.\nWith $s = \\sqrt{1 + z_1}$, the convergent binomial series, it is the union of the germs of the two smooth curves $\\set{z_2 = z_1 s}$ and $\\set{z_2 = -z_1 s}$, and neither contains the other.\nEvery germ of an analytic set has a decomposition $A_a = A_{1,a} \\cup \\cdots \\cup A_{k,a}$ into irreducible germs, unique up to order when no $A_{i,a}$ is contained in another.\n",
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    "name": "Irreducible analytic germ",
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