# How can I simplify $\mathbb{F}[X,Y]/(X^2-Y^2)$?

Here $$\mathbb{F}$$ is an arbitrary field. $$\psi :\mathbb{F}[X,Y]\rightarrow\mathbb{F}[X]\times\mathbb{F}[X];\\ f\mapsto (f \ \text{mod} (X+Y),f\ \text{mod} (X-Y))$$Is a ring homomorphism with kernel $$(X^2-Y^2)$$, so $$\mathbb{F}[X,Y]/(X^2-Y^2)$$ is isomorphic to a subring of $$\mathbb{F}[X]^2$$.

What is the form of this subring?

• $F[r,s]/(rs)$. Send $r=X+Y$ and $s=X-Y$. – user647486 Apr 7 at 11:04