Root-free exterior squares over finite fields

The paper is devoted to exterior squares of polynomials and matrices over the finite field for large q. We find the limit as d→∞ of the probability that a monic polynomial of degree d has root-free exterior square. We also find the limit as d→∞ of the probability that a matrix X GL(d,q) has eigenvalue-free exterior square. This should be useful in recognising GL(V) in its action on V V, when V is a d-dimensional vector space over .