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  "body": "\n## Definition\n\nLet $K$ be a field.\nAn *absolute value* on $K$ is a map $\\abs{\\cdot} \\colon K \\to \\R_{\\ge 0}$ such that for all $x, y \\in K$:\n\n1. $\\abs{x} = 0$ if and only if $x = 0$;\n2. $\\abs{xy} = \\abs{x}\\abs{y}$;\n3. $\\abs{x + y} \\le \\abs{x} + \\abs{y}$.\n\nIt is *trivial* if $\\abs{x} = 1$ for all $x \\ne 0$.\nTwo absolute values $\\abs{\\cdot}_1$ and $\\abs{\\cdot}_2$ are *equivalent* if there is a real $c \u003e 0$ with $\\abs{x}_2 = \\abs{x}_1^{c}$ for all $x \\in K$.\nA *place* of an [[algebraic number field|MWBDFS2KJ5]] $K$ is an equivalence class of nontrivial absolute values on $K$.\n\n## Remarks\n\nEquivalent absolute values define the same topology on $K$.\nA place is *infinite* (archimedean) if the values $\\abs{n \\cdot 1}$ for $n \\in \\N$ are unbounded, and *finite* (non-archimedean) otherwise.\n\nLet $K$ be a number field with $r_1$ real embeddings and $r_2$ pairs of complex embeddings.\nEach embedding $\\sigma \\colon K \\to \\C$ gives the absolute value $\\abs{x}_\\sigma = \\abs{\\sigma(x)}$, and conjugate embeddings give the same one.\nThese give the $r_1 + r_2$ infinite places.\nEach nonzero [[prime ideal|00MHHZ9SNE]] $\\mathfrak{p}$ of $\\mathcal{O}_K$ gives $\\abs{x}_{\\mathfrak{p}} = N(\\mathfrak{p})^{-v_{\\mathfrak{p}}(x)}$ for $x \\ne 0$.\nHere $v_{\\mathfrak{p}}(x)$ is the exponent of $\\mathfrak{p}$ in the factorisation of $x\\mathcal{O}_K$, and $N(\\mathfrak{p})$ is the [[norm|M54ADM0CVF]].\nThese give all finite places.\nEvery place arises in one of these two ways.\n\nExample: on $\\Q$, the [[p-adic absolute value|WGN82TCEF2]] $\\abs{x}_p = p^{-v_p(x)}$ satisfies the three clauses, even with $\\abs{x + y}_p \\le \\max(\\abs{x}_p, \\abs{y}_p)$, and it is nontrivial since $\\abs{p}_p = 1/p$; so it defines a place of $\\Q$.\nBy [[Ostrowski's theorem|GCQ23XT6VD]], the places of $\\Q$ are these and the place of the usual absolute value.\n\nNon-example: on $\\Q$, the map $x \\mapsto \\abs{x}^2$ fails clause 3, since $\\abs{1 + 1}^2 = 4 \u003e 1^2 + 1^2$.\nSo it is not an absolute value in this sense, although it is a power of one.\n",
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