I'm trying to show that R=Q[X,Y]/(Y2−X3)R=Q[X,Y]/(Y2−X3) is not a UFD, but I got stuck.
To prove this, I could try to find two "different" factorisations for one element, but I am not familiar with this, so I tried to use a lemma and one of the previous exercises. If my syllabus is right, in every UFD counts
xis irreducible⟺xis primex is irreducible⟺x is prime
My syllabus also states that the elements X¯,Y¯∈RX¯,Y¯∈R are irreducible. So I tried to show that at least one of the elements X¯,Y¯∈RX¯,Y¯∈R does not generate a prime ideal.
This would mean that I should find two polynomials f,g∈Q[X,Y]f,g∈Q[X,Y], such that
( 4 months ago )
As Jared notes, your approach is fine. You also consider directly the relation Y2−X3=0Y2−X3=0, which implies that Y2=X3Y2=X3. Does this give you any hints for an element which has two distinct factorizations?