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  "body": "\n## Summary\n\nA local field is a field with a nontrivial absolute value for which it is locally compact.\nThe Archimedean local fields are $\\R$ and $\\C$.\nThe others are the finite extensions of the fields $\\Q_p$ of [[p-adic numbers|BT6T702C28]] and the fields $\\F_q((t))$ of Laurent series over finite fields.\nThey are the completions of [[global fields|634XKY8RZ0]].\nNumber theory studies a global field one place at a time through its local fields.\n\n## Explanations\n\n### L1 human\n\nA local field is a field where one can measure size, where Cauchy sequences converge, and where closed balls are compact.\nThe real numbers are the most familiar example.\nThe $p$-adic numbers $\\Q_p$ are another: there, a ball such as $\\Z_p$ is compact, like a closed interval.\nStudying an equation in all local fields is a first test for whether it has rational solutions.\n",
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