Are most polynomials reducible or irreducible?
Let $k$ be an algebraically closed field and consider the ring $R = k[X_1, \ldots, X_n]$ of polynomials in $n$ variables over $k$.
Is the "general" polynomial in $R$ reducible or irreducible?
The exact meaning of the set of "general" polynomials is left to the pleasure of the reader, but two possibilities that come to mind are "outside a set of measure zero" (if $k = \mathbb C$ and w/r/t the Lebesgue measure) or "in the complement of a countable union of Zariski closed sets".
If $n = 1$ then any polynomial factors as a product of linear ones, so "most" polynomials are reducible. If $n > 1$ the opposite might be true, because by mumbling "Bertini theorem" the general member in a linear system corresponding to an ample divisor might be irreducible; since homogeneous polynomials in many variables are examples of such things (over projective space), there the general (homogeneous) polynomial could be irreducible.
Solution 1:
I agree with Matt E's comment that we should bound degrees. This makes the argument straightforward. The space of nonzero polynomials in $n$ variables of degree at most $d$ modulo scalar multiplication is a projective space of dimension ${d+n \choose n} - 1$. The subspace of all reducible polynomials is the union of the images in this projective space of $d - 1$ morphisms from products of lower-dimensional projective spaces taking the form
$$(f(x_1, ... x_n), g(x_1, ... x_n)) \mapsto f(x_1, ... x_n) g(x_1, ... x_n)$$
where $f$ has degree at most $d_1 \ge 1$, $g$ has degree at most $d_2 \ge 1$, and $d_1 + d_2 = d$. These products of projective spaces have dimension ${d_1 + n \choose n} + {d_2 + n \choose n} - 2$.
Claim: If $n \ge 2$ this dimension is always strictly less than ${d+n \choose n} - 1$.
Proof. The polynomial $f(d) = {d+n \choose n} - 1$ has no constant term and non-negative coefficients. The claim then follows from the observation that $(d_1 + d_2)^k > d_1^k + d_2^k$ when $k \ge 2, d_1, d_2 \ge 1$. $\Box$
It follows that the generic polynomial (in the complement of a finite union of Zariski closed subsets) of degree at most $d$ is irreducible for $n \ge 2$.