Affine Algebraic Sets in A^2 - Dummit and Foote, page 660, Example 3

[Math Amateur]

[I apologise for repeating this post ... but I genuinely would like help ... and the post comes from September 2015 ... and so is not an impatient "bump" of an item ... I hope administrators will understand ...]

=====================================================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 ...

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)):https://www.physicsforums.com/attachments/5664
In the above text, in Example (3), we find the following:

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

Peter====================================================

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:View attachment 5665
View attachment 5666
View attachment 5667

 

[Euge]

By definition $k[x,y] = (k[x])[y]$. So every polynomial in $k[x,y]$ is a polynomial in $y$ with coefficients in $k[x]$. The Euclidean algorithm over $k[x]$ yields $f(x,y) = (x^3 - y^2)g(x,y) + r(x,y)$ where $0 \le \operatorname{deg}_y(r(x,y)) < 2$ (note: $\operatorname{deg}_y p(x,y)$ denotes the degree of $p(x,y)$ as a polynomial in $y$). So $r(x,y) = f_0(x) + f_1(x)y$ for some $f_0(x), f_1(x)\in k[x]$.

 

[Math Amateur]

By definition $k[x,y] = (k[x])[y]$. So every polynomial in $k[x,y]$ is a polynomial in $y$ with coefficients in $k[x]$. The Euclidean algorithm over $k[x]$ yields $f(x,y) = (x^3 - y^2)g(x,y) + r(x,y)$ where $0 \le \operatorname{deg}_y(r(x,y)) < 2$ (note: $\operatorname{deg}_y p(x,y)$ denotes the degree of $p(x,y)$ as a polynomial in $y$). So $r(x,y) = f_0(x) + f_1(x)y$ for some $f_0(x), f_1(x)\in k[x]$.

Thanks so much for the help Euge ... much appreciated ...

Sorry for the late reply ... i have been ill with the flu ... just recovering ...

Thanks again,

Peter