Case 1: p $\not\!$| XYZ

First of all we see easily that (X, Y, Z) are not all congruent modulo p. If not, we have

3X $$\cong$$ X + Y + Z $$\cong$$ X^p + Y^p + Z^p $$\cong$$ 0(mod p)

Now, we are assuming that p $\geq$ 5 and so we obtain X $\cong$ 0(mod p); this contradicts our hypothesis for Case 1. Secondly, we see that (X + $\omega^{j}_{}$Y) are mutually co-prime in R as j runs over 0,..., p - 1. If not, then we have a prime ideal P in R containing (X + $\omega^{j}_{}$Y, X + $\omega^{k}_{}$Y). Then this ideal P contains (1 - $\omega^{j-k}_{}$)Y. Now from the factorisation

(- Z)^p = X^p + Y^p = (X + Y)(X + $$\omega$$Y) ^... (X + $$\omega^{p-1}_{}$$Y)

we see that P contains Z. Hence, by the assumption that (X, Y, Z) are mutually co-prime we see that P contains (1 - $\omega^{l}_{}$) for some 0 $\leq$ l $\leq$ p - 1. By the description of prime ideals in R as in section 1 we see that P = $\lambda$R. But then Z is a multiple of p which contradicts our hypothesis in Case 1. By the above paragraph and unique factorization of ideals we see that we have ideals I_j of R such that I_j^p = (X + $\omega^{j}_{}$Y)R. Assume I₁ is principal; then we have an equation

(X + $$\omega$$Y) = u ^. $$\alpha^{p}_{}$$

for some $\alpha$ $\in$ R and u a unit in R. Applying complex conjugation we obtain

(X + $$\omega^{-1}_{}$$Y) = $$\overline{u}$$ ^. $$\overline{\alpha}^{p}_{}$$

By the results mentioned in section 1 we have $\omega^{r}_{}$$\overline{u}$ = u for some r. Moreover, $\alpha^{p}_{}$ is congruent to an integer modulo pR and hence is congruent to its own complex conjugate. Thus we obtain an equation

X + $$\omega$$Y - $$\omega^{r}_{}$$X - $$\omega^{r-1}_{}$$Y $$\cong$$ 0(mod p)

Now it follows from the description of R given in Section 1 that it is a free abelian group with basis consisting of any (p - 1) elements of the set {1,$\omega$,...,$\omega^{p-1}_{}$}. From this and the fact that X and Y are prime to p it follows that r = 1 and X $\cong$ Y(mod p). By similar reasoning interchanging the roles of Y and Z we can conclude that there is an ideal J₁ such that J₁^p = (X + $\omega$Z). Assuming J₁ is principal we see by an argument like the one above that X $\cong$ Z(mod p). But as seen above the two congruences

X $$\cong$$ Y(mod p) andX $$\cong$$ Z(mod p)

contradict the hypothesis of Case 1. Hence, either I₁ or J₁ must be non-principal. But then by the principal result of Class Field theory as mentioned in section 1 we have required cyclic extension of K.