Chad Awtrey and recent alumni from Elon and Williams High School publish paper in algebra journal

The paper titled ""Galois groups of doubly even octic polynomials" appears in the most recent issue of the Journal of Algebra and Its Applications.

Associate Professor of Mathematics Chad Awtrey, Kiley Shannon ’18, Anna Altmann ’23, Sam Cryan, and Madeleine Touchette have published a research paper in the most recent issue of the Journal of Algebra and Its Applications.

The paper, “Galois groups of doubly even octic polynomials”, Journal of Algebra and Its Applications, Vol. 19, No. 1, 1-15, (2020), was based on work done during the 2017-18 academic year, when Shannon was finishing her honor’s thesis and Altmann, Cryan, and Touchette were students at Williams High School (WHS).

The collaboration was forged when Awtrey and Elon alumni Robin French ’15 and Dee Sizemore ’86, both of whom were teachers at WHS at the time, secured a grant from the Mathematical Association of America to provide enrichment activities for mathematically gifted students at WHS. One such activity was a research project, led by Awtrey, that produced this publication.

In their work, which was also presented at SURF in 2018, the authors studied a special class of polynomials, which they call “doubly even octic polynomials”, that are of the form x^8+ax^4+b where a and b are integers. Among the most important properties of such polynomials is the collection of symmetries of the polynomial’s roots; such symmetries encode all arithmetic properties of the polynomial. Building on previous work of even quartic polynomials (which are of the form x^4+ax^2+b), the authors discovered, in some circumstances, that simple algebraic relationships between a and b completely determine the collection of symmetries.

For example, one of their main results was the following: If b is not a perfect square and the number b(a^2-4b) is a perfect square, then the symmetries of the doubly even octic polynomial x^8+ax^4+b is always the same collection of 32 permutations of the roots. Since the collection of symmetries of a polynomial is, in general, very difficult to determine, the authors’ work provides an important contribution to the discipline by showing that such a determination can often be reduced to simple algebraic relationships among the coefficients. Moreover, their techniques for proving their results pave the way for future similar discoveries.

Altmann, Cryan and Touchette are children of Elon faculty; namely, Kyle Altmann (Associate Professor of Physics), Mark Cryan (Assistant Professor of Sport Management), and Brant Touchette (Professor of Biology and Environmental Studies). Anna Altmann ’23 is now a first-year student at Elon, Sam Cryan is a sophomore at Princeton University, and Madeleine Touchette is a first-year student at Xavier University. The other student coauthor, Kiley Shannon ’18, is working as a Certified Public Accountant at Ernst & Young in Seattle.

The grant from the Mathematical Association of America is ongoing. Awtrey managed the grant for the first two years, with assistance from Associate Professor of Mathematics Jim Beuerle and Lecturer in Statistics Ryne Vankrevelen. Currently, the grant activities are led by Beuerle and Vankrevelen.