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  "body": "\n## Definition\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]].\nA subset $B \\subseteq L$ is a *transcendence basis* of $L/K$ if:\n\n1. $B$ is [[algebraically independent|KV4PT6KW10]] over $K$;\n2. $L$ is an [[algebraic extension|09EYMKMSB9]] of the subfield $K(B)$ generated by $K$ and $B$.\n\n## Remarks\n\nEquivalently, $B$ is a maximal algebraically independent subset of $L$.\nExample: in the [[field of rational functions|4V81611P9N]] $K(x_1, \\dots, x_n)$, the set $\\set{x_1, \\dots, x_n}$ is a transcendence basis over $K$, and so is $\\set{x_1^2, x_2, \\dots, x_n}$.\nNon-example: $\\set{x_1}$ is not a transcendence basis of $K(x_1, x_2)/K$, because $x_2$ is not algebraic over $K(x_1)$, so clause 2 fails.\nFor an algebraic extension, the empty set is the unique transcendence basis.\n",
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