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  "body": "\n## Summary\n\nA set of elements of a field extension is algebraically independent if no nonzero polynomial relation with coefficients in the base field holds among them.\nIt is the analogue for polynomials of [[linear independence|0CRMGPAH83]] for linear combinations.\nA single element is algebraically independent exactly when it is [[transcendental|0NQMZ89X4A]].\nMaximal algebraically independent sets are [[transcendence bases|E45PA267S4]].\n\n## Explanations\n\n### L1 human\n\nTwo numbers are algebraically dependent when some polynomial equation ties them together.\nFor example, $\\pi$ and $\\pi^2$ are dependent over $\\Q$, since $y - x^2 = 0$ at $x = \\pi$, $y = \\pi^2$.\nIn the field $\\Q(x, y)$ of rational functions in two variables, $x$ and $y$ are independent: no nonzero polynomial in them is zero.\nWhether $\\pi$ and $e$ are algebraically independent over $\\Q$ is not known.\n",
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