So I was looking through past exam questions with a friend and we came across this one:
is irreducible over
My first thought was that it's obviously true because because I can't factor anything out, but I have no idea how to prove it rigorously enough for exam-standard.
I thought something along the lines of
Assuming it can be written as the product of a line and a lower dimensional curve. Then
but this leads to a contradiction.
Would I also need to consider the case where I can pull out some sort of conic as well?
You need more than just
is a UFD.
be a (nontrivial) factorisation. Then taking degrees in y, you have
. So we may assume WLOG
(since the presence of the y^3 term means you cannot have
--- in exam you need to expand that a little bit more).
Hence we may assume
, where p(x) divides x^4, so is one of
. Now examine each one in turn.