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  "body": "\n## Definition\n\nLet $K$ be a [[field|04C5ZBG1WT]] with an [[absolute value|0FFXKTKKQC]] $\\abs{\\cdot}$ that is nontrivial, that is, $\\abs{x} \\neq 1$ for some $x \\neq 0$.\nGive $K$ the topology of the metric $d(x, y) = \\abs{x - y}$.\nThen $K$ is a *local field* if $K$ is a [[locally compact space|0FE2SB8WVJ]] for this topology.\n\n## Remarks\n\nA local field is Archimedean when it is $\\R$ or $\\C$, and non-Archimedean otherwise.\nMany authors use \"local field\" only for the non-Archimedean case.\nA non-Archimedean field is a local field exactly when it is complete for a discrete valuation and its [[residue field|NJ3APQ8BHH]] is finite.\nThe non-Archimedean local fields of characteristic $0$ are the finite extensions of the fields $\\Q_p$.\nThose of characteristic $p$ are the fields $\\F_q((t))$ with $q$ a power of $p$.\nExample: $\\Q_p$ is a local field, because the ball $\\Z_p$ is a compact neighbourhood of $0$.\nNon-example: $\\Q$ with $\\abs{\\cdot}_p$ is not a local field, because it is not complete, and a field with such an absolute value is locally compact only when complete.\nNon-example: $\\C((t))$ is complete for a discrete valuation, but its residue field $\\C$ is infinite, so it is not locally compact.\n",
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