$k$ is a field whose characteristic is not $2$, and $f(x,y)=x^2-y^2, g(x,y)=x^2+x^3-y^2$.
Exercise. Show that $k[[x,y]]/(f)\simeq k[[x,y]]/(g)$.
So far, I've shown the following.
- $k[[x,y]]$ is an integral domain.
- $\sqrt{1+x}\in k[[x,y]]^{\times}$, $g(x,y)=(x\sqrt{1+x}+y)(x\sqrt{1+x}-y)$
- The two ideals $(x+y)$, $(x-y)$ are not coprime.
I was trying to use the Chinese remainder theorem. But since the two ideals are not coprime, it cannot help. Now I'm trying to use the fundamental homomorphism theorem.
If you can answer this exercise, please help me.