Chad Awtrey and math major publish in international algebra journal

The article by the associate professor of mathematics and mathematics major Ashley Pritchard ‘20 appeared in the International Journal of Algebra.

Associate Professor of Mathematics Chad Awtrey and mathematics major Ashley Pritchard ‘20 have published a paper in the most recent issue of the International Journal of Algebra.

Associate Professor of Mathematics Chad Awtrey

In the paper, “Totally Ramified Degree-p Extensions Over the Unramified Quadratic p-Adic Field”, International Journal of Algebra, 13, no. 7, 297-306, (2019), the authors develop both theory and computational results related to classifying arithmetic properties of certain polynomials with p-adic coefficients whose degrees are twice the odd prime number p. This extends previous work by S. Amano in 1971, who studied the related case of p-adic polynomials whose degrees are an odd prime number p.

For a prime number p, the p-adic numbers are essentially infinite strings of numbers whose digits range from 0 to p-1. These numbers have their own notion of geometry, which is different from the traditional Euclidean geometry that is studied in pre-college mathematics courses. As such, they have found applications in many diverse areas. In physics, there are applications to the p-adic theory of strings, quantum mechanics, and spin glasses. In the theory of dynamical systems, there are applications to ergodicity, structures of cycles/attractors, and cryptographic applications like p-adic stream ciphers. In biology, there are applications to genetics, molecular motors, and cognitive models.

In their paper, Awtrey and Pritchard gave a formula for the total number of distinct p-adic polynomials of degree twice the odd prime number p; this number is known to be finite by a result of M. Krasner from the 1960s. They went on to determine further arithmetic characteristics of distinct classes of polynomials. These invariants are interesting in their own right, but they also aid in the computation of the symmetries of polynomial roots, which play an important role in the research area of computational algebra. As an application to their newly-developed theory, Awtrey and Pritchard were able to compute a complete description of all symmetry properties of degree 22 polynomials with 11-adic coefficients. This is a notoriously difficult task, and their work extends previous results, which were only known up to degree 14.