| This thesis begins by giving a brief time line of the origins of Number Theory. It highlights the big theorems that have been constructed in this subject, along with the mathematicians who constructed them. The thesis, then, goes on to prove the Law of Quadratic Reciprocity for the Jacobi symbol. This includes proving Eisenstein's Lemma for the Jacobi symbol. Then, it is shown that Gauss's Lemma has an even greater generalization than Eisenstein's Lemma. Finally, this thesis shows the similarities between the rational integers and the Gaussian integers, including proving the Law of Quadratic Reciprocity for the Gaussian integers and constructing a similar version of Gauss's Lemma for the Gaussian integers. |
