Case 2: p| XYZ

We may assume that Z = p^kZ₀ and (p, X, Y, Z₀) are mutually co-prime. By writing p = (unit) ^. $\lambda^{(p-1)}_{}$ in the ring R, we obtain an equation of the form

U^p + V^p + (unit)$$\lambda^{mp}_{}$$W^p = 0 withm > 0

where (U, V, W) are in R so that (U, V, W,$\lambda$) are mutually co-prime. Let (U, V, W) be a collection of elements of R that satisfy such an equation with m the least possible. Then $\lambda$ divides one of the factors (U + $\omega^{j}_{}$V). But then we have

(U + $$\omega^{j}_{}$$V) - (U + $$\omega^{k}_{}$$V) = $$\omega^{j}_{}$$(1 - $$\omega^{k-j}_{}$$)V = (unit) ^. $$\lambda$$V

and thus, $\lambda$ divides all the factors (U + $\omega^{j}_{}$V). Moreover, since V is co-prime to p and thus $\lambda$ as well, we see that (U + $\omega^{j}_{}$V)/$\lambda$ have distinct residue classes modulo $\lambda$R. But then, by the pigeon-hole principle there is at least one 0 $\leq$ j $\leq$ (p - 1) such that (U + $\omega^{j}_{}$V) is divisible by $\lambda^{2}_{}$ in R. Replacing V by $\omega^{j}_{}$V we may assume that (U + V) is divisible by $\lambda^{l}_{}$ for some l > 1. Hence we may write

$$\lambda^{l}_{}$$ U + V =

$$\omega^{k}_{} \lambda$$ =

where all the a_k are elements of R that are co-prime to $\lambda$ and with each other (as in the previous case). This gives us the identity l + (p - 1) = mp or equivalently l = (m - 1)p + 1. Since l $\geq$ 2 we have m $\geq$ 2. Now by unique factorisation of ideals in R we see that there are ideals I_j in R such that I_j^p = a_jR. Assume that I₀, I₁ and I_{p - 1} are principal, then we have the equations

$$\lambda^{l}_{}$$ U + V =

$$\omega \lambda$$ =

$$\omega^{-1}_{} \lambda$$ =

for some units u, v and w in R and some elements b₀, b₁ and b_-1 in R. Eliminating U and V from these equations we obtain

$$\lambda^{l}_{}$$ ^. u ^. b₀^p - $$\lambda$$ ^. v ^. b₁^p = $$\omega$$($$\lambda$$ ^. w ^. b_-1^p - $$\lambda^{l}_{}$$ ^. u ^. b₀^p)

which becomes

b₁^p + v₁ ^. b_-1^p + $$\lambda^{l-1}_{}$$ ^. v₂b₀^p = 0

where v₁ and v₂ are units (we use here the fact that 1 + $\omega$ is a unit in R). Modulo pR the last term on the left-hand side vanishes since l $\geq$ p > (p - 1). Thus we see that v₁ is congruent to a p-th power and thus an integer modulo pR. By section 1 we have a representation of Galois as required, unless v₁ is a p-th power. If v₁ = v₃^p, then (U, V, W) = (b₁, v₃b_-1, b₀) satisfy

U^p + V^p + (unit)$$\lambda^{(m-1)p}_{}$$W^p = 0

which contradicts the minimality of m since we have seen that m $\geq$ 2. Thus, either we have constructed a cyclic extension of the required type or one of I₀, I₁, I_{p - 1} is non-principal. But then again by the principal result of Class Field theory we have a cyclic extension as required.