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  "body": "\n## Definition\n\nLet $B$ be a [[commutative ring|0310REJJY8]] and let $A$ be a [[subring|0GNCRPYXYV]] of $B$.\nAn element $b \\in B$ is *integral over $A$* if there are an integer $n \\ge 1$ and elements $a_0, \\ldots, a_{n-1} \\in A$ such that\n$$b^n + a_{n-1} b^{n-1} + \\cdots + a_1 b + a_0 = 0.$$\nEquivalently, $b$ is a root of a [[monic polynomial|0C3QBCR46M]] in $A[x]$.\nMore generally, for a ring homomorphism $f \\colon A \\to B$, an element $b$ is integral over $A$ if it is integral over the subring $f(A)$.\n\n## Remarks\n\nExample: $i \\in \\C$ is integral over $\\Z$, because $i^2 + 1 = 0$ and $x^2 + 1$ is monic with integer coefficients.\n\nNon-example: $1/2 \\in \\Q$ is not integral over $\\Z$.\nBy the [[rational root theorem|0819CQ1V8H]], a rational root of a monic polynomial with integer coefficients is an integer.\n\nThe following are equivalent for $b \\in B$:\n\n1. $b$ is integral over $A$.\n2. The subring $A[b]$ is a [[finitely generated|05QY1VEBYD]] $A$-module.\n3. $A[b]$ is contained in a subring $C$ of $B$ that is a finitely generated $A$-module.\n\nThe proof of 3 implies 1 uses the [[determinant trick|A8HRT9H3XA]].\nWhen $A$ and $B$ are fields, an element is integral exactly when it is [[algebraic|0H628RSGWX]], since a nonzero polynomial over a field can be divided by its leading coefficient.\n",
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