[ **Irreducibility of the Cayley-Menger determinant,**]{}\
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[Mowaffaq Hajja$^{1}$, Mostafa Hayajneh$^2$, Bach Nguyen$^3$, and Shadi Shaqaqha$^4$]{}\
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(1): P. O. Box 1, Philadelphia University, 19392, Amman, Jordan\
mowhajja@yahoo.com\
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(2), (4): Yarmouk University, Irbid, Jordan\
hayaj86@yahoo.com, shadish2@yahoo.com\
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(3): Louisiana State University, Baton Rouge, LA, USA\
bnguy38@lsu.edu

> [**Abstract.**]{} [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 