Kutztown University of Pennsylvania Deptartment of Mathematics

Colloquium

3:30 p.m. Tuesday, February 16, 2010

Lytle Hall 228

The Randy Johnsons of Abstract Algebra and Fields of Dreams

Dr. Eric Landquist

Assistant Professor of Mathematics Kutztown University of Pennsylvania

Abstract

With the football season officially over, many anticipate the beginning of baseball. As a tribute to recently retired pitcher Randy Johnson, we will discuss big units. We are familiar with the concept of a unit: ±1, ±i, or anything with an absolute value of 1. We will generalize this idea to define units of algebraic number fields such as ℚ (\sqrt{D}) = \{a + b\sqrt{D} | a, b ∈ ℚ\}, where D ∈ ℤ, our first

field of dreams. These units are found by solving the Pell Equation: x^{2} − Dy^{2} = 1. Surprisingly, units aren’t always easy to find in these situations; they can be very big. One method that we will illustrate uses continued fraction expansions of \sqrt{D}. This idea will motivate us to describe an Abelian group-like algebraic structure called the infrastructure of the field, and will allow us to speed up the computation of these units. In the case that the units are too big to even store on a computer, we compute the natural logarithm of the unit, which is called the regulator of the field. Finally, we will apply these ideas to fields of rational functions modulo a prime, our second field of dreams. The benefit of these so-called function fields is that the regulator is now an integer and also the order of a group related to the units and infrastructure of the field. Furthermore, the infrastructure has applications to cryptography, hence our interest in the problem. All are welcome to attend; this talk will be accessible to students and faculty alike!

3:00 p.m. refreshments served

3:30 p.m. talk begins

Kutztown Univ. of Pennsylvania Dept. of Mathematics Dr. Paul S. Ache, III, Chair