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  "body": "\n## Definition\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \\in L$.\nThe element $a$ is transcendental over $K$ if $a$ is not [[algebraic|0NPW1DWECM]] over $K$.\nThat is, $f(a) \\neq 0$ for every nonzero $f \\in K[x]$.\n\n## Remarks\n\nThe number $\\pi$ is transcendental over $\\mathbb{Q}$, but it is algebraic over $\\mathbb{R}$, as a root of $x - \\pi$.\nThe variable $x$ is transcendental over $K$ in the field $K(x)$ of rational functions.\nIf $a$ is transcendental over $K$, then $K(a)$ is isomorphic to $K(x)$, and $[K(a) : K]$ is infinite.\n\n## Explanations\n\n### L1 human\n\nA transcendental element satisfies no polynomial equation over the smaller field.\nIt behaves like a free variable.\nThe transcendental numbers are the complex numbers that are transcendental over $\\mathbb{Q}$, such as $\\pi$ and $e$.\n",
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