Graduate Student

Event Information How to Not Solve an Equation
18:10 on Thursday October 27, 2016
19:00 on Thursday October 27, 2016
BA6183, Bahen Center, 40 St. George St.
Zackary Wolske

University of Toronto

Cubic number fields, and in general cubic rings, have a nice parametrization in terms of degree three homogeneous binary forms. Some questions about the structure of a number field can be answered by looking at a corresponding binary form. For example, suppose we want to know if the ring of integers in a cubic number field has a power basis, i.e. when can we write every integral element as $a+b\alpha+c\alpha^2$, where $a,b$ and $c$ are rational integers. This turns out to be equivalent to asking if the binary form represents 1. For a certain set of forms, we can reduce this to a question on Mordell's equation: for which $k$ does $x^3=y^2+k$ have integer solutions? I will introduce the relevant background, then give the necessary restrictions for solving this algebraically, and explain why they fail to work in this case.