This "proof" is actually a pretty good manifestation of the current limitations of GPT3; it mangles together two different proofs in a way that does not make sense.
It starts by setting up for an elementary proof (a/b = sqrt(2) with a, b minimal implies 2|a^2 implies 2|a implies 4|a^2 implies 2|b^2 implies 2|b which is a contradiction). But instead of following through it finishes using another proof based on the rational root theorem, and makes up some bullshit on the way to make the argument flow nicely (it does not follow from a^2 / b^2 = 2 with no common factors that 2 is a perfect square).
It starts by setting up for an elementary proof (a/b = sqrt(2) with a, b minimal implies 2|a^2 implies 2|a implies 4|a^2 implies 2|b^2 implies 2|b which is a contradiction). But instead of following through it finishes using another proof based on the rational root theorem, and makes up some bullshit on the way to make the argument flow nicely (it does not follow from a^2 / b^2 = 2 with no common factors that 2 is a perfect square).