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.

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