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  "body": "\n## Definition\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\\mathcal{O}_K$, and let $\\mathfrak{a}$ be a nonzero [[ideal|096Z9MP2ME]] of $\\mathcal{O}_K$.\nThe *norm* of $\\mathfrak{a}$ is the [[cardinality|02CK6QMXYD]] of the [[quotient ring|0076HWHGHQ]]:\n$$N(\\mathfrak{a}) = \\#(\\mathcal{O}_K / \\mathfrak{a}).$$\nThis number is finite.\n\n## Remarks\n\nExample: in $\\Z[i]$, the ideal $(1 + i)$ has the classes of $0$ and $1$, so $N((1 + i)) = 2$, which equals the [[norm of the Gaussian integer|0HYRJF4WNQ]] $1 + i$, namely $1^2 + 1^2$.\nNon-example: the zero ideal is excluded, since $\\mathcal{O}_K / (0) \\cong \\mathcal{O}_K$ is infinite.\nThe norm is multiplicative: $N(\\mathfrak{a}\\mathfrak{b}) = N(\\mathfrak{a}) N(\\mathfrak{b})$.\nFor a nonzero $\\alpha \\in \\mathcal{O}_K$, $N(\\alpha \\mathcal{O}_K) = \\abs{N_{K/\\Q}(\\alpha)}$, the absolute value of the field norm.\nBy multiplicativity the norm extends to a group homomorphism from the nonzero [[fractional ideals|YB4JTB4WJN]] of $\\mathcal{O}_K$ to $\\Q_{\u003e0}^{\\times}$.\nFor a nonzero prime ideal $\\mathfrak{p}$ lying over the prime $p$, $N(\\mathfrak{p}) = p^f$, where $f$ is the residue degree.\n",
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