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

## TL;DR

This paper investigates the existence of zero divisors of support size three in group algebras over finite fields, contributing to the broader conjecture about zero divisors in torsion-free group algebras.

## Contribution

It provides a detailed analysis of potential length 3 zero divisors in rational and finite field group algebras, advancing understanding of the conjecture.

## Key findings

- Zero divisors of support size 2 cannot exist.
- The study characterizes possible length 3 zero divisors.
- Results inform the ongoing effort to prove the absence of zero divisors in torsion-free group algebras.

## 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 $p$.