Associate Professor of Mathematics Chad Awtrey and alumna Briana Brady ‘19, who earned her degree in mathematics, have published a paper in the most recent issue of the JP Journal of Algebra, Number Theory, and Applications.

Associate Professor of Mathematics Chad Awtrey and alumna Briana Brady ‘19 have published a paper in the most recent issue of the JP Journal of Algebra, Number Theory, and Applications.

In their paper, “Automorphisms of 2-adic Fields of Degree Twice an Odd Number”, JP Journal of Algebra, Number Theory, and Applications, **44**, no. 2, 201-210, (2019), Awtrey and Brady generalize previous results in the field of *p*-adic number theory concerning arithmetic properties of polynomials with 2-adic coefficients.

To understand the context of this work, we need to consider binary numbers, which are strings of 0s and 1s. Since there are only two possibilities for each digit of a binary number, we say a binary number is a “base 2” representation of a number. Not only do binary numbers play an important role in how information is stored on a computer, but they also have many fascinating applications in number theory and cryptography, since the representation of numbers in different bases reveals important properties for that number. Expanding the collection of binary numbers to include infinite strings of 0s and 1s yields the set of 2-adic numbers. One area of current mathematical research focuses on studying arithmetic properties of polynomials with 2-adic coefficients, where the goal is to understand how many such “distinct” polynomials exist and whether defining properties of the distinct classes of polynomials can be systematically characterized.

The most important classes of polynomials with 2-adic coefficients are the so-called Eisenstein polynomials, which simply have the property that every coefficient is a multiple of two and the constant term is not a multiple of four. A classical result from the last century states that there are 6 distinct Eisenstein degree two polynomials with 2-adic coefficients. Recent research from J. Jones and D. Roberts in 2006 has shown that the number of distinct Eisenstein polynomials of degree six is 30, and for degree ten the number is 126. Recent work from Awtrey and Elon alumna Erin Strosnider ‘14 showed that this number is 510 when the degree is fourteen.

Awtrey and Brady were able to generalize these results, in several ways. For example, if *n* is an odd number, they showed the number of distinct Eisenstein polynomials of degree *2n* is equal to *2^(n+2)-2*. Moreover, they determined several important characteristics of each polynomial, including a systematic representation of each distinct class of polynomials. Brady presented her work at Elon’s Spring Undergraduate Research Forum in April 2019, and she is currently working as an Implementation Consultant for Fast Enterprises in the state of Washington.