Zero divisors of support size $3$ in group algebras and trinomials divided by irreducible polynomials over $GF(2)$

Abstract

A famous conjecture about group algebras of torsion-free groups states that there is no zero divisor in such group algebras. A recent approach to settle the conjecture is to show the non-existence of zero divisors with respect to the length of possible ones, where by the length we mean the size of the support of an element of the group algebra. The case length $2$ cannot be happen. The first unsettled case is the existence of zero divisors of length $3$. Here we study possible length $3$ zero divisors in rational group algebras and in the group algebras over the field with $p$ elements for some prime $p$.