In this chapter, we fix an integer *n* ≥ 3 which is not a power of the characteristic *p*, and a monic polynomial *f*(*x*) *∈ k*[*x*] of degree *n, f*(*x*) = ∑^{n}_{i=0} *A _{i}x^{i}, A_{n}* = 1.

**Lemma 23.1.** *Suppose that f has n distinct roots in k, all of which are nonzero (i.e., A*_{0} ≠ 0*). Let χ be a nontrivial character of k ^{×} with χ^{n}* = .

*Proof.* Let *ρ* be a nontrivial character of *k ^{×}*. Then

det(*Frob*_{k, ρ}|*!*(*N*)) = ...

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