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  "body": "\n## Definition\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]].\nThe extension $L/K$ is *transcendental* if it is not [[algebraic|09EYMKMSB9]], that is, if some element of $L$ is [[transcendental|00SW8B91E7]] over $K$.\n\n## Remarks\n\nExample: $\\R/\\Q$ is transcendental, because $\\pi$ is a [[transcendental number|079EGSD32Y]].\nExample: for a field $K$, the extension $K(t)/K$ of rational functions is transcendental, since $g(t) \\ne 0$ for every nonzero $g \\in K[x]$.\nNon-example: $\\Q(\\sqrt 2)/\\Q$ is not transcendental, because it is finite and hence algebraic.\nEvery field extension $L/K$ is the composite of a purely transcendental step $K \\subseteq K(B)$ and an algebraic step $K(B) \\subseteq L$, where $B$ is a transcendence basis.\n",
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