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  "body": "\n## Definition\n\nA *global field* is a [[field|04C5ZBG1WT]] $K$ of one of the following two kinds.\n\n1. $K$ is a [[finite extension|0C1Y04SZ1W]] of the [[rational numbers|0NCJ5WXN6Z]] $\\Q$; then $K$ is a [[number field|MWBDFS2KJ5]].\n2. $K$ is a finite extension of the field $\\F_q(t)$ of rational functions in one variable $t$ over a [[finite field|0ZXZ82FTX2]] $\\F_q$. Then $K$ is a global function field.\n\n## Remarks\n\nExample: $\\Q(\\sqrt{2})$ is a global field, because it has degree $2$ over $\\Q$.\nExample: $\\F_3(t)$ itself is a global function field, of degree $1$ over $\\F_3(t)$.\nNon-example: $\\R$ is not a global field, because it has infinite degree over $\\Q$ and has characteristic $0$.\nNon-example: $\\C(t)$ is not a global field, because its field of constants $\\C$ is not finite.\nThe global function fields are exactly the function fields of smooth projective curves over finite fields.\n",
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