  We study the irreducibility of Wronskian Hermite polynomials labelled by
partitions. It is known that these polynomials factor as a power of x times a
remainder polynomial. We show that the remainder polynomial is irreducible for
the partitions (n, m) with m <= 2, and (n, n) when n + 1 is a square.
  Our main tools are two theorems that we prove for all partitions. The first
result gives a sharp upper bound for the slope of the edges of the Newton
polygon for the remainder polynomial. The second result is a Schur-type
congruence for Wronskian Hermite polynomials.
  We also explain how irreducibility determines the number of real zeros of
Wronskian Hermite polynomials, and prove Veselov's conjecture for partitions of
the form (n, k, k-1, ..., 1).
