The rings $ F[x,y]/ (y^{2} - x) $ and $F[x,y]/( y^{2} - x^{2}) $ are not isomorphic for any field F.
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2The second one has zero-divisors $[y-x]$ and $[y+x]$. The first one has none. – Henno Brandsma Sep 19 '20 at 08:16
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@HennoBrandsma thanks! – Learner_Shas Sep 19 '20 at 08:18
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1What have you tried? The first one is isomorphic to $F[y]$, see this post. – Dietrich Burde Sep 19 '20 at 08:22
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$F[x,y](y^2-x) \simeq F[y]$ (e.g. see here and its answer) and this has no-zero-divisors.
In $F[x,y](y^2-x^2)$ we have zero-divisors $\langle y - x \rangle$ and $\langle y + x \rangle$ (they multiply to $\langle y^2 - x^2 \rangle = 0$).
So the rings cannot be isomorphic.
Henno Brandsma
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