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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)):
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
====================================================
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:
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 (
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:
Attachments

D&F  1  Affine Algebraic Sets  Examples ... .png132.8 KB · Views: 747

D&F  1  Affine Algebraic Sets  Ch 15  Page 1.png120.4 KB · Views: 720

D&F  2  Affine Algebraic Sets  Ch 15  Page 2 ... .png111.6 KB · Views: 648

D&F  3  Affine Algebraic Sets  Ch 15  Page 3 ... .png111.6 KB · Views: 657