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

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