# Thread: Affine Algebraic Sets - D&F Chapter 15, Section 15.1 - Example 3 - page 660

1. ## Affine Algebraic Sets - D&F Chapter 15, Section 15.1 - Example 3 - page 660

I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, Example 3 on page 660 reads as follows: (see attachment)

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Let $\displaystyle V = \mathcal{Z}(x^3 - y^2)$ in $\displaystyle \ \ \mathbb{A}^2$.

If $\displaystyle (a, b) \in \mathbb{A}^2$ is an element of V, then $\displaystyle a^3 = b^2$.

If $\displaystyle a \ne 0$, then also $\displaystyle b \ne 0$ and we can write$\displaystyle a = (b/a)^2, \ b = (b/a)^3$.

It follows that V is the set $\displaystyle \{ (a^2, a^3) \ | \ a \in k \}$.

For any polynomial $\displaystyle f(x,y) \in k[x,y]$. we can write $\displaystyle f(x,y) = f_0(x) + f_1(x)y + (x^3 - y^2)g(x,y)$

... ... ... etc etc

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I cannot follow the line of reasoning:

"For any polynomial $\displaystyle f(x,y) \in k[x,y]$. we can write $\displaystyle f(x,y) = f_0(x) + f_1(x)y + (x^3 - y^2)g(x,y)$"

Can anyone clarify why this is true and why D&F are taking this step?

Peter