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  "body": "\n## Definition\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\\mathcal{O}_K$.\nFor a nonzero [[ideal|096Z9MP2ME]] $\\mathfrak{a}$ of $\\mathcal{O}_K$, let $N(\\mathfrak{a}) = \\abs{\\mathcal{O}_K/\\mathfrak{a}}$ be its [[norm|M54ADM0CVF]].\nLet $s$ be a complex variable with $\\operatorname{Re} s \u003e 1$.\nThe *Dedekind zeta function* of $K$ is\n$$\\zeta_K(s) = \\sum_{\\mathfrak{a}} \\frac{1}{N(\\mathfrak{a})^{s}},$$\nwhere $\\mathfrak{a}$ runs over the nonzero ideals of $\\mathcal{O}_K$.\n\n## Remarks\n\nThe series converges absolutely for $\\operatorname{Re} s \u003e 1$.\nUnique factorisation of ideals gives the Euler product $\\zeta_K(s) = \\prod_{\\mathfrak{p}} \\big(1 - N(\\mathfrak{p})^{-s}\\big)^{-1}$ over the nonzero prime ideals $\\mathfrak{p}$.\nThe function has an [[analytic continuation|0M3YQJ81NB]] to a meromorphic function on $\\C$ whose only pole is a simple pole at $s = 1$.\nLet $K$ have $r_1$ real embeddings, $r_2$ pairs of complex embeddings, and $w_K$ roots of unity.\nLet $h_K$ be its [[class number|KTKK9B1X41]], $R_K$ its [[regulator|RWK0HNF1NA]] and $d_K$ its [[discriminant|E0VZTNPJXA]].\nThe residue is\n$$\\operatorname{Res}_{s = 1} \\zeta_K(s) = \\frac{2^{r_1} (2\\pi)^{r_2} h_K R_K}{w_K \\sqrt{\\abs{d_K}}},$$\nthe [[class number formula|55XX5A7XZV]].\n\nExample: for $K = \\Q$ every nonzero ideal is $n\\Z$ for a unique $n \\ge 1$, with norm $n$, so $\\zeta_{\\Q}$ is the [[Riemann zeta function|1XWBY4N0VT]].\nFor $K = \\Q(i)$, comparing Euler factors prime by prime gives $\\zeta_K(s) = \\zeta(s) L(s, \\chi_4)$, where $L(s, \\chi_4)$ is the [[Dirichlet L-function|9D2Z543EJQ]] of the non-trivial character modulo $4$.\n\nNon-example: the sum $\\sum_{\\alpha} \\abs{N_{K/\\Q}(\\alpha)}^{-s}$ over nonzero elements $\\alpha \\in \\mathcal{O}_K$ is not $\\zeta_K(s)$.\nIt counts each principal ideal once for each unit, so for $K = \\Q(\\sqrt{2})$, which has infinitely many units, it diverges for every $s$.\n",
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