I have a polynomial of degree 8 in 6 variables given explicitly by


$$ (\sqrt{1+(x\_1+x\_2+x\_3)^2+(y\_1+y\_2+y\_3)^2}+\sqrt{1+x\_1^2+y\_1^2}+\sqrt{1+x\_2^2+y\_2^2}+\sqrt{1+x\_3^2+y\_3^2})\times\text{the other seven of its Galois conjugates}. $$


I fed it into Maple and it shows it's irreducible. But being unsure of what is going on behind the scene, I am asking here for a less computation-intensive and more conceptual proof of this fact, in the sense that any computation involved in the proof can be checked by hand in a reasonable amount of time.



Check irreducibility of an explicit polynomial, without computer
For new variables $x$ and $y$ set $x\_1=x\_2=x\_3=x$ and $y\_1=y$, $y\_2=2y$, $y\_3=3y$. Then
$$
\prod\left(\sqrt{1+9x^2+36y^2}\pm\sqrt{1+x^2+y^2}\pm\sqrt{1+x^2+4y^2}\pm\sqrt{1+x^2+9y^2}\right)=-1024\left((3y^2 + 4)x^6 + (23y^4 + 47y^2)x^4 + (45y^6 + 180y^4)x^2 + 225y^6\right).
$$
This polynomial has degree $8$, and if it is irreducible, then the original polynomial is irreducible even more. Replacing $x$ with $xy$ and dividing by $y^6$ afterwards reduces to show that
$$
(3x^6 + 23x^4 + 45x^2)y^2 + 4x^6 + 47x^4 + 180x^2 + 225
$$
is irreducible. That holds, because the coefficients of $y^0$ and $y^2$ are relatively prime and non-squares.


