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 the rational group algebras and in the group algebras over the field ${\mathbb{F}}_p$ with $p$ elements for some prime $p$. As a consequence we prove that the rational group algebras of torsion-free groups which are residually finite $p$-group for some prime $p\ne 3$ have no zero divisor of length 3. We note that the determination of all zero divisors of length 3 in group algebras over ${\mathbb{F}}_2$ of cyclic groups is equivalent to find all trinomials (polynomials with 3 non-zero terms) divided by irreducible polynomials over ${\mathbb{F}}_2$. The latter is a subject studied in coding theory and we add here some results, e.g. we show that $1+x+x^2$ is a zero divisor in the group algebra over ${\mathbb{F}}_2$ for some element $x$ of the group if and only if $x$ is of finite order divided by $3$ and we find all $\beta$ in the group algebra of the shortest length such that $(1+x+x^2)\beta =0$; and $1+x^2+x^3$ or $1+x+x^3$ is a zero divisor in the group algebra over ${\mathbb{F}}_2$ for some element $x$ of the group if and only if $x$ is of finite order divided by 7.