Yep, I'm trying to understand the basics of the proof for my Extended Essay(An IB thing), as I am looking into what extent infinite descent(which is commonly used to prove specific cases) impacted the final result(which actually doesn't use infinite descent in its general proof).

Yeah, unless you're really good at abstract algebra, don't do that (furthermore, proving things is probably the most boring thing to do in maths: if you know it works, you know it works, no proof needed).