  In this paper we resolve the Alon-Jaeger-Tarsi conjecture for sufficiently
large primes. Namely, we show that for any finite field $\mathbb{F}$ of size
$61<|\mathbb F|\ne 79$ and any nonsingular matrix $M$ over $\mathbb{F}$ there
exists a vector $x$ such that neither $x$ nor $Ax$ has a 0 component.
