Let μ be a probability measure on ℤ that is not a Dirac mass and that has finite support. We prove that if the coefficients of a monic polynomial f(x) ∈ Z[x] of degree n are chosen independently at random according to μ while ensuring that f(0) ≠ 0 , then there is a positive constant θ= θ(μ) such that f(x) has no divisors of degree ≤ θn with probability that tends to 1 as n→ ∞ . Furthermore, in certain cases, we show that a random polynomial f(x) with f(0) ≠ 0 is irreducible with probability tending to 1 as n→ ∞ . In particular, this is the case if μ is the uniform measure on a set of at least 35 consecutive integers, or on a subset of [− H, H] ∩ Z of cardinality ≥ H4 / 5(log H) 2 with H sufficiently large. In addition, in all of these settings, we show that the Galois group of f(x) is either An or Sn with high probability. Finally, when μ is the uniform measure on a finite arithmetic progression of at least two elements, we prove a random polynomial f(x) as above is irreducible with probability ≥ δ for some constant δ= δ(μ) > 0 . In fact, if the arithmetic progression has step 1, we prove the stronger result that the Galois group of f(x) is An or Sn with probability ≥ δ .