Certifying Irreducibility in [Formula: see text]

We consider the question of certifying that a polynomial in [Formula: see text] or [Formula: see text] is irreducible. Knowing that a polynomial is irreducible lets us recognise that a quotient ring is actually a field extension (equiv. that a polynomial ideal is maximal). Checking that a polynomial is irreducible by factorizing it is unsatisfactory because it requires trusting a relatively large and complicated program (whose correctness cannot easily be verified). We present a practical method for generating certificates of irreducibility which can be verified by relatively simple computations; we assume that primes and irreducibles in [Formula: see text] are self-certifying.