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) = ∑ni=0 Aixi, An = 1.
Lemma 23.1. Suppose that f has n distinct roots in k, all of which are nonzero (i.e., A0 ≠ 0). Let χ be a nontrivial character of k× with χn = . Form the object N := Lχ(f)(1/2) ∈ Parith. Its Tannakian determinant “det” (N) is geometrically of finite order. It is geometrically isomorphic to δa for a = (–1)n A0 = (–1)n f(0) = the product of all the zeroes of f.
Proof. Let ρ be a nontrivial character of k×. Then ρ is good for N, and
det(Frobk, ρ|!(N)) = ...