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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 algebraic over $K$ if there is a nonzero polynomial $f \\in K[x]$ with $f(a) = 0$.\n\n## Remarks\n\nThe number $\\sqrt{2}$ is algebraic over $\\mathbb{Q}$, because it is a [[root|00EHX6FWC0]] of $x^2 - 2$.\nEvery element $a \\in K$ is algebraic over $K$, as a root of $x - a$.\nThe element $a$ is algebraic over $K$ if and only if $[K(a) : K]$ is finite.\nAn element that is not algebraic is [[transcendental|00SW8B91E7]].\n\n## Explanations\n\n### L1 human\n\nAn algebraic element solves a polynomial equation with coefficients in the smaller field.\nThe algebraic numbers are the complex numbers that are algebraic over $\\mathbb{Q}$.\n",
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