Separable exterior squares over finite fields

The paper concerns exterior squares of polynomials and matrices over the finite field for large q. We find the probability that monic has a non-separable exterior square. We then find the probability that X GL(d,q) has an exterior square which is non-separable, non-cyclic or non-semisimple. This should have applications in recognising GL(V) in its action on V V, when V is a d-dimensional vector space over .