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  "body": "\n## Definition\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $S \\subseteq L$.\nThe set $S$ is *algebraically independent over $K$* if, for every $n \\ge 1$, all distinct $s_1, \\dots, s_n \\in S$ and every nonzero polynomial $f \\in K[x_1, \\dots, x_n]$, one has $f(s_1, \\dots, s_n) \\ne 0$.\nOtherwise $S$ is *algebraically dependent over $K$*.\n\n## Remarks\n\nEquivalently, the evaluation map from the [[polynomial ring|0Q9EZS90GY]] $K[x_s : s \\in S]$ to $L$ that sends $x_s$ to $s$ is injective.\nA one-element set $\\set{s}$ is algebraically independent if and only if $s$ is [[transcendental|00SW8B91E7]] over $K$.\nExample: in $K(x, y)$, the set $\\set{x, y}$ is independent, since a polynomial $f(x, y)$ that is zero as an element of $K(x, y)$ is the zero polynomial.\nExample: $\\set{\\pi}$ is independent over $\\Q$, because $\\pi$ is a [[transcendental number|079EGSD32Y]].\nNon-example: $\\set{\\pi, \\pi^2}$ is dependent over $\\Q$, since $f(x_1, x_2) = x_2 - x_1^2$ is nonzero and vanishes at $(\\pi, \\pi^2)$.\nAlgebraic independence implies [[linear independence|0EVHJFTG9Z]] over $K$, by taking $f$ linear; the converse fails, as $\\set{1, \\pi}$ shows.\n",
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