In this essay, we see how cyclotomic fields can lead to elegant proofs of
number theoretical concepts. We will prove some elementary properties of prime
cyclotomic fields (a cyclotomic field obtained by adjoining a primitive p-th
root of unity to Q, where p is an odd prime), and use them to prove the laws of
Quadratic and Cubic Reciprocity. We will also explore the applications of
cyclotomic fields in certain forms of Diophantine equations.