How do you prove the irreducibility of this multivariate polynomial?

Question

Let $p(x,y,z) = (x-r)^{2a} + (y-s)^{2b} + (z-t)^{2c}\in\Bbb Q[x,y,z]$ be a polynomial for some $a,b,c \geq 1$. How do you prove that it's irreducible over $\Bbb{Q}$?

$r,s,t \in \Bbb{Q}, \neq 0$.

Answer 1

First of all, the polynomial is not always irreducible (imagine $a=b=c.$) Second, I think a good way is to substitute values of $y, z$ and use Selmer's criterion for irreducibility of (univariate) trinomials. I haven't gone through the details, but on quick glance it seems to work.