An Introduction to p-adic Numbers
Daniel Ford
Boston University
We begin by reviewing the construction of the real numbers, highlighting a rather naive and uninteresting choice involved in their realization. We can then overcome this naivety and construct the p-adic numbers themselves. The p-adic numbers which are also rational can be thought of as a “base p expansion”, as opposed to the typical fractions of integers. The implications of p-adic numbers include the amazing proof of Fermat’s Last Theorem developed by Sir Andrew Wiles in 1995. While Wiles’ proof lays outside the realm of understanding for most mathematicians, an interesting question called the Local-Global Principle is well within reach, and we will discuss some of its marvelous answers.
Daniel Ford graduated from Providence College as a mathematics major in 2010. He is currently a fourth-year graduate student and Teaching Fellow at Boston University. He is studying number theory, which in its most basic form studies properties of the integers and extensions thereof. Beyond his doctoral studies, he works with a great program called PROMYS for Teachers, which is a 6-week summer course in elementary number theory for high school mathematics teachers. The program is steeped in history and strives to combine higher mathematics with the basic high school education.
