Selected exercises from

Proof. First, we show that F [x, y]/(y−x) ∼= F [t]. Let φ : F [x, y]→ F [t] be an homomorphism given by φ(x) = t, φ(y) = t and φ(f) = f (for all f ∈ F ). Note that the kernel of the homomorphism is precisely (y−x), therefore by the 1st isomorphism theorem F [x, y]/(y−x) ∼= F [t]. Furthermore since F [t] is an integral domain, so is F [x, y]/(y − x). Now, we will show that F [x, y]/(y − x) has zero divisors, thus is not an integral domain and the conclusion will follow. Observe that (y − x)(y + x) = y − x = 0. We are done as neither (y − x) nor (y + x) is zero.