Proof: We show something a bit stronger, that F[x,y]/(yn−x)≇F[x,y]/(x2−y2) for any n, with F only an integral domain.
First we show y−x,y+x∉(y2−x2)=(y−x)(y+x) so that there are zero divisors in F[x,y]/(y2−x2). Assuming the contrary we have for f some polynomial y−x=(y−x)(y+x)f⇒(y+x)f=1, an impossibility. Similarly for y+x∉(y2−x2).
Now we show F[x,y]/(yn−x)=F[y]/(yn−x)≅F[y] is an integral domain, which will complete the proof. To see the first equality, notice that ⊇ is clear and any polynomial of F[x,y]/(yn−x) can be represented by the image of a polynomial devoid of variables in x since ¯yn=¯x. Letting φ be the natural surjective homomorphism of F[y] onto F[y]/(yn−x), we can observe that if ker φ≠0 then 0≠(yn−x)f∈F[y] for some f∈F[x,y], an impossibility by (yn−x)f=ynf−xf and observing the result of the xf term by choosing the result of the multiplication monomial with maximal degree of x in f, which cannot be affected by any terms in xf (else the f monomials' y and x degrees are the same and can be rewritten as one) nor in ynf. ◻
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