Lemma 10.161.10. Let R be a Noetherian normal domain with fraction field K of characteristic p > 0. Let a \in K be an element such that there exists a derivation D : R \to R with D(a) \not= 0. Then the integral closure of R in L = K[x]/(x^ p - a) is finite over R.
Proof. After replacing x by fx and a by f^ pa for some f \in R we may assume a \in R. Hence also D(a) \in R. We will show by induction on i \leq p - 1 that if
is integral over R, then D(a)^ i a_ j \in R. Thus the integral closure is contained in the finite R-module with basis D(a)^{-p + 1}x^ j, j = 0, \ldots , p - 1. Since R is Noetherian this proves the lemma.
If i = 0, then y = a_0 is integral over R if and only if a_0 \in R and the statement is true. Suppose the statement holds for some i < p - 1 and suppose that
is integral over R. Then
is an element of R (as it is in K and integral over R). Applying D we obtain
is in R. Hence it follows that
is integral over R. By induction we find D(a)^{i + 1}a_ j \in R for j = 1, \ldots , i + 1. (Here we use that 1, \ldots , i + 1 are invertible.) Hence D(a)^{i + 1}a_0 is also in R because it is the difference of y and \sum _{j > 0} D(a)^{i + 1}a_ jx^ j which are integral over R (since x is integral over R as a \in R). \square
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