Example 10.147.3. Let p be a prime number. The ring extension
R = \mathbf{Z}[1/p] \subset R' = \mathbf{Z}[1/p][x]/(x^{p - 1} + \ldots + x + 1)
has the following property: For d < p there exist elements \alpha _0, \ldots , \alpha _{d - 1} \in R' such that
\prod \nolimits _{0 \leq i < j < d} (\alpha _ i - \alpha _ j)
is a unit in R'. Namely, take \alpha _ i equal to the class of x^ i in R' for i = 0, \ldots , p - 1. Then we have
T^ p - 1 = \prod \nolimits _{i = 0, \ldots , p - 1} (T - \alpha _ i)
in R'[T]. Namely, the ring \mathbf{Q}[x]/(x^{p - 1} + \ldots + x + 1) is a field because the cyclotomic polynomial x^{p - 1} + \ldots + x + 1 is irreducible over \mathbf{Q} and the \alpha _ i are pairwise distinct roots of T^ p - 1, whence the equality. Taking derivatives on both sides and substituting T = \alpha _ i we obtain
p \alpha _ i^{p - 1} = (\alpha _ i - \alpha _1) \ldots \widehat{(\alpha _ i - \alpha _ i)} \ldots (\alpha _ i - \alpha _1)
and we see this is invertible in R'.
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