  If $S$ is a given regular $n$-simplex, $n \ge 2$, of edge length $a$, then
the distances $a_1$, $\cdots$, $a_{n+1}$ of an arbitrary point in its affine
hull to its vertices are related by the fairly known elegant relation
$\phi_{n+1} (a,a_1,\cdots,a_{n+1})=0$, where $$\phi = \phi_t (x,
x_1,\cdots,x_{n+1}) = \left( x^2+x_1^2+\cdots+x_{n+1}^2\right)^2 - t\left(
x^4+x_1^4+\cdots+x_{n+1}^4\right).$$ The natural question whether this is
essentially the only relation is answered positively by M. Hajja, M. Hayajneh,
B. Nguyen, and Sh. Shaqaqha in a recently submitted paper entitled "Distances
from the vertices of a regular simplex." In that paper, the authors made use of
the irreducibility of the polynomial $\phi $ in the case when $n \ge 2$,
$t=n+1$, $x= a \ne 0$, and $k = \mathbb{R}$, but supplied no proof, promising
to do so in another paper that is turning out to be this one. It is thus the
main aim of this paper to establish that irreducibility. In fact, we treat the
irreducibility of $\phi$ without restrictions on $t$, $x$, $a$, and $k$. As a
by-product, we obtain new proofs of results pertaining to the irreducibility of
the general Cayley-Menger determinant that are more general than those
established by C. D'Andrea and M. Sombra in Sib. J. Math. 46, 71--76.
