I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ...(adsbygoogle = window.adsbygoogle || []).push({});

At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ...

I need someone to help me to fully understand the reasoning/analysis behind one of the statements in Example (3) on Page 660 of D&F ...

On page 660 (in Section 15.1) of D&F we find the following text and examples (I am specifically focused on Example (3)):

In the above text, in Example (3), we find the following:

"... ... For any polynomial ##f(x,y) \in k[x,y]## we can write

##f(x,y) = f_0(x) + f_1(x)y + (x^3 - y^2) g(x,y).##"

Can someone explain ( slowly and carefully) exactly how/why this is true ... ...

Peter

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In order for readers of the above post to understand the context of the question and the notation employed I am providing the introductory pages on affine algebraic sets in the D&F text ... ... as follows:

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# I Affine Algebraic Sets - Dummit and Foote, page 660, Ex. 3

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