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

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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 $?

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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