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  "body": "\n## Summary\n\nAn atomic domain is an [[integral domain|09S3PFBEPM]] in which every nonzero nonunit is a finite product of [[irreducible elements|052SDEPZH7]].\nIt asks for the existence of factorizations, but not for their uniqueness.\nEvery Noetherian domain is atomic, and more generally every domain with the ascending chain condition on principal ideals.\nA [[unique factorisation domain|06YMXJMXYS]] is exactly an atomic domain in which every irreducible element is [[prime|0MNHXBSX3M]].\n\n## Explanations\n\n### L1 human\n\nThe irreducible elements are the \"atoms\" of multiplication: they cannot be split into two nonunit factors.\nIn an atomic domain, every nonzero nonunit can be broken down into atoms in at least one way.\nIn $\\Z[\\sqrt{-5}]$ this works, but the breakdown is not unique: $6 = 2 \\cdot 3 = (1 + \\sqrt{-5})(1 - \\sqrt{-5})$.\nIn the ring of all algebraic integers it fails completely: every nonzero nonunit $\\alpha$ is $\\sqrt{\\alpha} \\cdot \\sqrt{\\alpha}$, so there are no atoms at all.\n",
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