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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]].\nAn *absolute value* on $K$ is a map $\\abs{\\cdot} \\colon K \\to \\R$ such that:\n\n1. $\\abs{x} \\ge 0$ for all $x \\in K$, and $\\abs{x} = 0$ if and only if $x = 0$.\n2. $\\abs{xy} = \\abs{x} \\abs{y}$ for all $x, y \\in K$.\n3. $\\abs{x + y} \\le \\abs{x} + \\abs{y}$ for all $x, y \\in K$.\n\nA *normed field* is a field $K$ together with an absolute value on $K$.\nIt is a metric space with the distance $d(x, y) = \\abs{x - y}$.\n\n## Remarks\n\nExample: $\\C$ with the modulus $\\abs{a + bi} = \\sqrt{a^2 + b^2}$ is a normed field; clause 2 is $\\abs{zw} = \\abs{z}\\abs{w}$, and clause 3 is the triangle inequality in the plane.\nExample: $\\Q$ with the [[p-adic absolute value|WGN82TCEF2]] for a prime $p$ is a normed field.\nExample: every field with the trivial absolute value, $\\abs{x} = 1$ for $x \\ne 0$, is a normed field.\nNon-example: on $\\R$, the map $x \\mapsto x^2$ satisfies clauses 1 and 2 but fails clause 3, since $(1 + 1)^2 = 4 \u003e 2 = 1^2 + 1^2$.\nClauses 1 and 2 give $\\abs{1} = 1$ and $\\abs{-x} = \\abs{x}$.\nAn absolute value that satisfies the stronger inequality $\\abs{x + y} \\le \\max(\\abs{x}, \\abs{y})$ is a [[non-Archimedean absolute value|1H5F1BP6XS]].\nA normed field is a [[topological field|D7YQW9RG24]] for the topology of its metric.\n",
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