Chad Awtrey and alumnus publish research on symmetries of polynomial roots

The paper titled "Galois groups of even sextic polynomials" appears in the most recent issue of the Canadian Mathematical Bulletin.

Associate Professor of Mathematics Chad Awtrey and Peter Jakes ’17 have published a research paper in the most recent issue of the Canadian Mathematical Bulletin.

The paper, “Galois groups of even sextic polynomials”, Canadian Mathematical Bulletin, Vol. 63, No. 3, 670-676 (2020) discusses new theory and algorithmic methods to determine the symmetry properties of polynomials of the form x^6+ax^4+bx^2+c, which they call even sextic polynomials. For such a polynomial, the collection of symmetries is known as the polynomial’s Galois group and is named in honor of French mathematician Evariste Galois (1811-1832). The properties of the polynomial’s Galois group govern many important arithmetic properties of the polynomial’s roots. Therefore, a fundamental task in computational algebra is to determine the structure of a polynomial’s Galois group.

For a general degree 6 polynomial, there are 16 possibilities for the structure of its Galois group. However, only 8 of these groups actually occur as the Galois group of an even sextic polynomial. In their work, Awtrey and Jakes give a very fast algorithm for determining the structure of an even sextic polynomial’s Galois group. As an application, they provide new one-parameter families of polynomials for each possible Galois group. An implementation of their algorithm is available on Awtrey’s website.