Exercise 8, § 1

Commutative Algebra, IX, p. 43

Let $A$ be a ring, and $\xi$ an element of $A$ such that $\sum_{i=0}^{p-1} \xi^i = 0$ . Then, in $W(A)$ , one has the equation

V_A(1) = \sum_{i=0}^{p-1} \tau_A(\xi^i).

Deduce that if

A

is of characteristic

p

, one has

V_A \circ F_A(a) = p.a \quad \text{for all } a \in W(A).

Solution

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