Summary: We look at the construction of a particular subclass of cyclic codes known as the Bose-Chaudhuri-Hocquenghem (BCH) Codes. These codes are constructed with a prescribed minimum distance, which means that the codes can be designed to correct as many errors as are required for the intended application. Our goal is to construct classes of BCH codes in as simple a fashion as possible by using minimal sets of generators. An elementary upper bound on the number of generators necessary for our construction is fairly easy to obtain. Our main result shows that for sufficiently large length, the upper bound is actually sharp. Along the way, we provide a host of results on the structure of cyclotomic cosets, along with an introduction to algebraic coding theory.