  Let $\mu$ be a probability measure on $\mathbb{Z}$ that is not a Dirac mass
and that has finite support. We prove that if the coefficients of a monic
polynomial $f(x)\in\mathbb{Z}[x]$ of degree $n$ are chosen independently at
random according to $\mu$ while ensuring that $f(0)\neq0$, then there is a
positive constant $\theta=\theta(\mu)$ such that $f(x)$ has no divisors of
degree $\le \theta n$ with probability that tends to 1 as $n\to\infty$.
  Furthermore, in certain cases, we show that a random polynomial $f(x)$ with
$f(0)\neq0$ is irreducible with probability tending to 1 as $n\to\infty$. In
particular, this is the case if $\mu$ is the uniform measure on a set of at
least 35 consecutive integers, or on a subset of $[-H,H]\cap\mathbb{Z}$ of
cardinality $\ge H^{4/5}(\log H)^2$ with $H$ sufficiently large. In addition,
in all of these settings, we show that the Galois group of $f(x)$ is either
$\mathcal{A}_n$ or $\mathcal{S}_n$ with high probability.
  Finally, when $\mu$ 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 $\ge\delta$ for some constant
$\delta=\delta(\mu)>0$. In fact, if the arithmetic progression has step 1, we
prove the stronger result that the Galois group of $f(x)$ is $\mathcal{A}_n$ or
$\mathcal{S}_n$ with probability $\ge\delta$.
