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If $S$ is a given regular $n$-simplex, $n \ge 2$, of edge length $a$, thenthe distances $a_1$, $\cdots$, $a_{n+1}$ of an arbitrary point in its affinehull 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 isessentially the only relation is answered positively by M. Hajja, M. Hayajneh,B. Nguyen, and Sh. Shaqaqha in a recently submitted paper entitled "Distancesfrom the vertices of a regular simplex." In that paper, the authors made use ofthe 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, promisingto do so in another paper that is turning out to be this one. It is thus themain aim of this paper to establish that irreducibility. In fact, we treat theirreducibility of $\phi$ without restrictions on $t$, $x$, $a$, and $k$. As aby-product, we obtain new proofs of results pertaining to the irreducibility ofthe general Cayley-Menger determinant that are more general than thoseestablished by C. D'Andrea and M. Sombra in Sib. J. Math. 46, 71--76.

Irreducibility of the Cayley-Menger determinant, and of a class of related polynomials