# Chad Awtrey publishes number theory research with math major and alumni

Associate Professor of Mathematics Chad Awtrey, math major Peter Komlofske '19, and alumni Christian Reese '18 and Janae Williams '18 have published their article "On Galois p-adic fields of p-power degree" in the most recent issue of the JP Journal of Algebra, Number Theory, and Applications.

Prime numbers, polynomials, and symmetries are the main objects of study in a recently published research paper written by Associate Professor of Mathematics Chad Awtrey and his undergraduate researchers Peter Komlofske '19, Christian Reese '18, and Janae Williams '18. Associate Professor of Mathematics Chad AwtreyAppearing in the most recent issue of the JP Journal of Algebra, Number Theory, and Applications (volume 41, issue 2, pages 275-287), the authors' article "On Galois p-adic fields of p-power degree" adds to the growing body of pure mathematics literature surrounding classifications of distinct classes of polynomials whose roots exhibit what are known as "cyclic" symmetries when viewed through a prime number lens.

It is common to study properties of numbers by writing them in different bases.   If the final digit of the representation is 0, we know immediately the number is divisibly by that base. For example, the decimal number 40 can be written as 101000 in base 2 (also known as binary), 130 in base 5, and 37 in base 11. This shows 40 is divisible by 2 and 5 but not 11.

Similarly, it is natural to study arithmetic properties of roots of polynomials in terms of different "polynomial bases". These properties are encoded in what mathematicians call "extension fields". For a given extension field, there are many ways of representing it by polynomials; just as there are many ways to represent a decimal number in different bases.

If we are interested in prime number properties of these roots, then we can isolate our attention on one fixed prime number and ask, how many different extensions fields exist whose prime-number properties are fundamentally different?  Research from the 1960s shows that this number is finite, and so mathematicians are interested in completely classifying various properties of these extension fields, such as: the number of different fields, one polynomial that defines each field, and symmetry properties of that polynomial's roots.

The paper by Awtrey, Komlofske, Reese, and Williams focuses on polynomials whose degree is a power of the prime number p.  Their main result gives exactly one polynomial per field extension of degree p^2 and p^3 when the symmetries form the rotations of a regular polygon whose number of sides equals the degree of the polynomial. Their classification extends previous work from the 1970s that focused on the case where the degree was equal to the prime number p.