  We consider the question of certifying that a polynomial in ${\mathbb Z}[x]$
or ${\mathbb Q}[x]$ 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 ${\mathbb F}_p[x]$ are self-certifying.
