Just a suggestion: you might be able to give some kind of geometric argument. If you decompose , then it would have to be into linear terms, which are hyperplanes. You could try to claim that your surface cannot be the product of hyperplanes.
I was wondering if anyone could think of an elegant proof that the polynomial
is irreducible in .
(My problem had , but I don't think it's relevant for this part.)
I have one proof, but it's basically brute force, using that if then (WLOG) , and just writing out what the factors look like. I was wondering if there was a... nicer way.