(1.4) then becomes ( b$_{k}$$_{0}$$_{+}$$_{t}$ + 2 $^{p}$− 1 2 ∑ k =1 b$_{k}$ ) x ≡ − ( (2 − $^{1}$t mod p ) b$_{k}$$_{0}$ + $^{p}$− 2 2 ∑ k =1 tb$_{k}$ ) ( p ). It is clear that c ≡ − 2 − $^{1}$t ( p ) is the solution to this congruence. By its uniqueness, we see that u is primary ⇔ t ≡ 0 ( p ) ⇔ u = ζ $^{t}$u is real. □ 2. Application to a Special Case of Fermat’s Last Theorem Fermat’s well-known final theorem, proved by Andrew Wiles and Richard Taylor in 1994, states that x n + y n = z n where x, y, z, n ∈ Z has no non-trivial solutions ( x, y, z ) for n ≥ 3. In fact, to prove this theorem, it suffices to prove that x $^{p}$+ y p = z p has no integral solutions for any positive odd prime p , since x n 0 + y n 0 = z n 0 ⇒ x p 1 + y p 1 = z p 1 where p is an odd prime dividing n (exists since n ≥ 3) and ( x$_{1}$, y$_{1}$, z$_{1}$ ) = ( x n/p 0 , y n/p 0 , z n/p 0 ). In other words, we can restrict our study to the case where n is an odd prime. There is a very elegant proof of a special case of this theorem using cyclotomy. The main use of the concept here is that it allows us to transform a ”sum of n -th powers” problem into a ”divisibility” problem since we can now factor x p + y p as $^{p}$− 1 ∏ i =0 ( x + ζ i $_{p}$y ). In this section, we shall lay out said proof. Let K = Q ( ζ ) where ζ = ζ$_{p}$ . We will suppose that for some ( x$_{0}$$_{,}$y$_{0}$, z$_{0}$ ) is a solution to x p + y p = z p for some odd prime p . Then (2.1) x p 0 + y p 0 = z p 0 WLOG, we can take x$_{0}$ , y$_{0}$ and z$_{0}$ to be pairwise relatively prime, for if some d ∈ Z divides two of them, it must divide the 3 $^{rd}$, and then x p 0 + y p 0 = z p 0 ⇔ x p 1 + y p 1 = z p 1 where x$_{0}$, y$_{0}$, z$_{0}$ = dx$_{1}$, dy$_{1}$, dz$_{1}$ respectively, with x$_{1}$, y$_{1}$, z$_{1}$ ∈ Z . We shall now reduce the problem to a special case and suppose that p does not divide the class number h of O$_{K}$ , and that p ∤ x$_{0}$y$_{0}$z$_{0}$ . From (2.1), we shall reach a contradiction. This case has been treated in Number Theory textbooks such as [1]. However, using the ON THE APPLICATIONS OF CYCLOTOMIC FIELDS IN INTRODUCTORY NUMBER THEORY 5