### Abstract:

One way to construct the real numbers involves creating equivalence classes
of Cauchy sequences of rational numbers with respect to the usual absolute
value. But, with a different absolute value we construct a completely different
set of numbers called the p-adic numbers, and denoted Qp. First, we take
an intuitive approach to discussing Qp by building the p-adic version of p7.
Then, we take a more rigorous approach and introduce this unusual p-adic
absolute value, j jp, on the rationals to the lay the foundations for rigor in Qp.
Before starting the construction of Qp, we arrive at the surprising result that
all triangles are isosceles under j jp. Then, we quickly construct Qp and extend
j jp from the rationals. Next, we leave equivalence classes of Cauchy sequences
behind and introduce a more understandable view of numbers in Qp. With this
view, we compute some p-adic numbers and observe that these computations are
similar to analogous computations in the real numbers. Then, we end our tour
of Qp with a proof of Henselnulls Lemma - a result describing a general approach
to building p-adic numbers. Lastly, we move to fi nite fi eld extensions of Qp. We
extend j jp to these field extensions with the help of the norm function, and end
the paper with two important propositions that characterize most finite fi eld
extensions of Qp.