Certifying Irreducibility in Z[x]

Abstract

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.

11 pages, 0 figures

References in corpus (1)

• Bounds on Factors in Z[x]