Time: 9:30 - 11:00, Wednesday March 23, 2022

Abstract: Let n be a positive integer. We show that if the equation

(1)     $x^4 + 2^n y^4 =z^4$

has a solution $(x, y, z)$ in a cubic number field $K$ with $xyz
eq 0$, then the Galois group of the field $K$ is the symmetric group $S_3$. In addition, we show that for every positive integer $d > 1$, there exists a number field $K_d$ of degree $d$ such that equation (1) has a solution $(x, y, z)$ in $K_d$ with $xyz
eq 0$. This paper extends the recent work of Bremner and Choudhry.

Online: https://meet.google.com/esi-huxm-xqg