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The Stacks project

Lemma 10.38.6. Let R \subset S be an inclusion of domains. Assume R is normal. Let g \in S be integral over R. Then the minimal polynomial of g has coefficients in R.

Proof. Let P = x^ m + b_{m-1} x^{m-1} + \ldots + b_0 be a polynomial with coefficients in R such that P(g) = 0. Let Q = x^ n + a_{n-1}x^{n-1} + \ldots + a_0 be the minimal polynomial for g over the fraction field K of R. Then Q divides P in K[x]. By Lemma 10.38.5 we see the a_ i are integral over R. Since R is normal this means they are in R. \square


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