Prime Numbers

204 Chapter 4 PRIMALITY PROVING
fact. The key fact that we will need is that K is the homomorphic image of
the ring Zp[x]/(x r − 1) where the coset representing x is sent to ζ. Indeed,all
that is used for this observation is that h(x)|x r − 1. Let G denote the image
of G under this homomorphism. Thus,
G = {γ ∈ K : γ = g(ζ) forsomeg(x) ∈ G}.
Note that if g(x) ∈ G and j ∈ J, theng(ζ) j = g(ζ j ).
Let d denote the order of the subgroup of Z ∗ r generated by n and p. Let
Gd = {g(x) ∈ G : g(x) =0or degg(x) n √ d .
Recall that K ∼ = Fp[x]/(h(x)), where h(x) is an irreducible polynomial in
Fp[x]. Denote the degree of h(x) byk. Thus,K ∼ = F p k, so it follows that if
j, j0 arepositiveintegerswithj ≡ j0 (mod p k −1), and β ∈ K, thenβ j = β j0 .
Let
J ′ = {j ∈ Z : j>0, j ≡ j0 (mod p k − 1) for some j0 ∈ J}.