ON THE APPLICATIONS OF CYCLOTOMIC FIELDS IN INTRODUCTORY NUMBER THEORY KABALAN GASPARD Abstract. In this essay, we see how prime cyclotomic fields (cyclotomic fields obtained by adjoining a primitive p -th root of unity to Q , where p is an odd prime) can lead to elegant proofs of number theoretical concepts. We namely develop the notion of primary units in a cyclotomic field, demonstrate their equivalence to real units in this case, and show how this leads to a proof of a special case of Fermat’s Last Theorem. We finally modernize Dirichlet’s solution to Pell’s Equation. Throughout this paper, unless specified otherwise, ζ ≡ ζ$_{p}$ ≡ e 2 $_{π}$√ − 1 p where p is an odd prime. K ≡ Q ( ζ ) and O K is the ring of integers of K . We assume knowledge of the basic properties of prime cyclotomic fields that can be found in any introductory algebraic number theory textbook, namely that: We also assume elementary knowledge of quadratic characters, quadratic reciprocity, and the Legendre symbol ( k p ) . Definition 1. Let α ∈ O K with α prime to p . Then α is primary iff α is congruent to a rational integer modulo (1 − ζ$_{p}$ ) $^{2}$. Date : June 22, 2011, re-edited February 11, 2012. 1 1. Primary elements in O K arXiv:1110.4445v3 [math.NT] 29 May 2012 • Gal ( K : Q ) ≃ U ( Z /p Z ) (the group of units of Z /p Z ), which is cyclic and of order p − 1. • O K = Z [ ζ$_{p}$ ] = 〈 1 , ζ$_{p}$, ..., ζ $^{p}$− 2 p 〉 $_{Z}$, where { 1 , ζ$_{p}$, ..., ζ $^{p}$− 2 p } is a Z -basis for O $_{K}$. The only roots of unity in (i.e. solutions in C to x n = 1 for some n • The only roots of unity in O K (i.e. solutions in C to x = 1 for some n ∈ N ) are of the form ± ζ i $_{p}$, i ∈ Z .