The study of solutions to diophantine equations has motivated research in many fields of mathematics, and was fundamental in the development of the famous class group associated to a number field. Though finite and abelian, this group, which in some sense captures the arithmetic of our field, is still not well understood. We will focus our attention to the "easiest" number fields, namely quadratics, and discuss a theorem which describes the 2-part (sylow subgroup) of the class group. If time permits, we will explore generalizations to cubic fields. (No prior knowledge of the class group will be assumed, and everything discussed will be framed in the hopes of answering questions like, Are there integer solutions to the equation y^2+5=x^3?)