as indeterminates is irreducible over ${\mathbb{R}}$ when $a \ne 0$ and $n \ge 2$.

The main goal of this paper is to prove this, and to actually give a complete treatment of the irreducibility of the more general polynomial $$\begin{aligned}
 \label{g}
g &=& \left(a^2+x_1^2+\cdots+x_{n+1}^2\right)^2 - t\left(a^4+x_1^4+\cdots+x_{n+1}^4\right) \in k[x_1,\cdots,x_{n+1}]\end{aligned}$$ for any field $k$, any $t \in k$, and any $a \in k$ (including the case $a=0$). It turns out that if ${\mbox{char~}}k \ne 2$, then $g$ is irreducible except in the two cases $(n,a,t)=(2,0,2)$ and $(n,a,t)=(2,0,3)$. When $(n,a,t)=(2,0,2)$, $g$ is the Heron polynomial with the well known factorization given in (\[uuu\]). When $(n,a,t)=(2,0,3)$, $g$ is reducible if and only if $k$ contains a primitive third root of unity, in which case $g$ factors as in (\[vvvv\]). If ${\mbox{char~}}k = 2$, then $g$ reduces to the polynomial $(1-t)(a+x_1+\cdots+x_{n+1})^4.$

The cases when $n=0$ and $n=1$ are ignored because they have no geometric significance. Also, the case $n=1$ turned out to be rather lengthy and complex, but it gives rise to interesting number theoretic aspects. We do not include these cases in this paper.

The polynomial obtained from $g$ by replacing $n+1$ with $n$ and $t$ with $n$ is essentially the Cayley-Menger determinant of 