Ailon and Rudnick have shown that if a,b∈C[T] are multiplicatively independent polynomials, then deg(gcd(aⁿ-1,bⁿ-1)) is bounded for all n≧1. We show that if instead a,b∈F[T] for a finite field F of characteristic p, then deg(gcd(aⁿ-1,bⁿ-1)) is larger than Cn for a constant C=C(a,b)>0 and for infinitely many n, even if n is restricted in various reasonable ways (e.g., p\notdivide n).