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

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    Super Member Bernhard's Avatar
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    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)

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

    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
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    Last edited by Bernhard; Oct 27th 2013 at 01:20 AM.
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