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  "body": "\n## Summary\n\nLet $\\mathfrak{a}$ be a nonzero ideal in the ring of integers $\\mathcal{O}_K$ of a [[number field|MWBDFS2KJ5]].\nThe norm of $\\mathfrak{a}$ is the number of elements of the quotient ring $\\mathcal{O}_K / \\mathfrak{a}$.\nIt is multiplicative, and for a principal ideal it is the absolute value of the field norm of a generator.\nIt measures the size of an ideal.\nIt is used in the [[finiteness of the class number|F99RM1T02C]] and in the definition of the [[Dedekind zeta function|KXS5DQDFYD]].\n\n## Explanations\n\n### L1 human\n\nIn the integers, the ideal $n\\Z$ has $\\abs{n}$ residue classes, so its size is $\\abs{n}$.\nThe ideal norm extends this to other number rings: it counts the residue classes modulo the ideal.\nFor example, in the Gaussian integers $\\Z[i]$ the ideal generated by $1 + i$ has two classes, so its norm is $2$, and indeed $(1 + i)(1 - i) = 2$.\nA smaller ideal has more residue classes: $4\\Z$ lies inside $2\\Z$ and has norm $4$ instead of $2$.\n",
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