In [4], Carlitz showed that if $n = k(p - 1)p^r$ where $0 < k < p$, then
$B_n(x)$ is irreducible (always with respect to {\bf Q} unless otherwise
indicated), namely that $pB_n(x)$ is $p$-Eisenstein.  He also showed that
if $2m + 1 = k(p - 1) + 1$ where $p$ is an odd prime and $1 \le k \le p$,
then $B_{2m + 1} / x(x - {1 \over 2})(x - 1)$ has an irreducible factor of
degree $\ge 2m + 1 - p$.
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