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  "body": "\n## Statement\n\nLet $L/K$ be a [[finite extension|0NRYPM9Y27]].\nThen $L/K$ is [[algebraic|09EYMKMSB9]].\n\n## Remarks\n\nThe proof: let $n = [L : K]$ and $a \\in L$. The $n + 1$ elements $1, a, \\dots, a^n$ are linearly dependent over $K$. A linear dependence $c_0 + c_1 a + \\dots + c_n a^n = 0$ gives a nonzero polynomial with root $a$.\nSo every element of $L$ has degree at most $[L : K]$ over $K$.\nThe converse is false, as the algebraic numbers show.\nAn extension generated by finitely many algebraic elements is finite, by the [[tower law|0A7KJ68XHT]].\n\n## Explanations\n\n### L1 human\n\nIn a finite-dimensional extension, the powers of an element cannot all be independent.\nA dependence between the powers is a polynomial equation for the element.\n",
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