# Thread: Algebraic Geometry - D&F Section 15.1, Exercose 24

1. ## Algebraic Geometry - D&F Section 15.1, Exercose 24

Dummit and Foote Section 15.1, Exercise 24 reads as follows:

---------------------------------------------------------------------------------------------------------

Let $\displaystyle V = \mathcal{Z} (xy - z) \subseteq \mathbb{A}^3$.

Prove that V is isomorphic to $\displaystyle \mathbb{A}^2$

and provide an explicit isomorphism $\displaystyle \phi$ and associated k-algebra isomorphism $\displaystyle \widetilde{\phi}$ from $\displaystyle k[V]$ to $\displaystyle k[ \mathbb{A}^2]$ along with their inverses.

Is $\displaystyle V = \mathcal{Z} (xy - z^2)$ isomorphic to $\displaystyle \mathbb{A}^2$?

-------------------------------------------------------------------------------------------------------------

I would appreciate some help and guidance with getting started with this exercise [I suspect I might need considerable guidance! :-( ]

Some of the background and definitions are given in the attachment.

Peter