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  "body": "\n## Statement\n\nLet $K \\subseteq L \\subseteq M$ be fields.\nIf $M/L$ and $L/K$ are [[algebraic|09EYMKMSB9]], then $M/K$ is algebraic.\n\n## Remarks\n\nThe proof: let $a \\in M$ have minimal polynomial $x^n + c_{n-1} x^{n-1} + \\dots + c_0$ over $L$. The field $E = K(c_0, \\dots, c_{n-1})$ is finite over $K$, and $E(a)$ is finite over $E$. By the [[tower law|0A7KJ68XHT]], $E(a)/K$ is finite, so $a$ is algebraic over $K$.\nAlso, if $M/K$ is algebraic, then $M/L$ and $L/K$ are algebraic.\n\n## Explanations\n\n### L1 human\n\nAlgebraic extensions can be stacked.\nIf each step adds only algebraic elements, then the whole tower adds only algebraic elements.\n",
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