But if the coefficients are integers, you can use NTT with a big enough modulus and get exact results and a boost (esp. in hardware) in multiplication time.
Sure, pick a large prime. Double it and add 1 (and call it n). If it's still prime, then you know the prime factorization of n-1. Pick your generator, and check if it raised to the p is 1, or if squaring it is one. If not, it's a generator of the multiplicative group mod n.