Let $X:=V(x^m-y^n)$ be a subspace of $\mathbb{A}^2$. How can I prove that if $(n,m)=1$ then $X$ is irreducible?
I think that it is isomorphic to $\mathbb{P}^1$ but I can't prove that.
Let $X:=V(x^m-y^n)$ be a subspace of $\mathbb{A}^2$. How can I prove that if $(n,m)=1$ then $X$ is irreducible?
I think that it is isomorphic to $\mathbb{P}^1$ but I can't prove that.
To prove that it is irreducible, show that it is the image of an irreducible space under a continuous map.
For example, you should have no trouble finding a map $\mathbb A^1\to\mathbb A^2$ whose image is your $X$.