In 1928, the Artin-Hasse Exponential E(x) was created and itâ\euro™s considered an analogue of the exponential function that comes from infinite products. It also has applications in formal group schemes and is studied in the p-adic number system. In this paper, fundamental results about the field of the p-adic rationals, Qp, like completion, are proven while smaller propositions are left to the reader. The integrality of E(x) is shown using Dworkâ\euro™s Lemma and extensions of the Artin Hasse exponential are further discussed.