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  "body": "\n## Statement\n\nLet $A \\subseteq B \\subseteq C$ be commutative rings.\nIf $B$ is [[integral|B7X0KFY1S1]] over $A$ and $C$ is integral over $B$, then $C$ is integral over $A$.\n\n## Remarks\n\nLet $c \\in C$ and take a monic relation $c^n + b_{n-1} c^{n-1} + \\cdots + b_0 = 0$ with $b_i \\in B$.\nEach $b_i$ is [[integral|1EHN4KPSFS]] over $A$, so $A' = A[b_0, \\ldots, b_{n-1}]$ is a [[finitely generated|05QY1VEBYD]] $A$-module.\nThe relation shows that $A'[c]$ is a finitely generated $A'$-module, so $A'[c]$ is a finitely generated $A$-module.\nHence $c$ is integral over $A$ by the module criterion for integrality.\n\nExample: $\\sqrt[4]{2}$ is integral over $\\Z[\\sqrt{2}]$, as a root of $x^2 - \\sqrt{2}$, and $\\Z[\\sqrt{2}]$ is integral over $\\Z$.\nSo $\\sqrt[4]{2}$ is integral over $\\Z$; indeed it is a root of $x^4 - 2$.\n\n## Explanations\n\n### L1 human\n\nIf each step of a tower of rings only adds roots of monic equations, then the whole tower does too.\nThis mirrors the fact that an algebraic extension of an algebraic extension of fields is algebraic.\n\n### L2 human\n\nThe proof does not build the monic polynomial over $A$ directly; it passes through finite generation of modules.\nThe same argument shows that the elements of $B$ integral over $A$ form a subring, the [[integral closure|G0V86SJBQW]] of $A$ in $B$.\nTransitivity then shows that the integral closure is integrally closed in $B$.\n",
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