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

[Math Amateur]

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)):View attachment 4749In 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 why this is true ... ...

Peter

 

[jvicens]

, thank you for your question. Let's take a closer look at the statement in Example (3) and break it down step by step.

First, let's define some terms. In this context, k is a field and k[x,y] represents the ring of polynomials in two variables x and y with coefficients in k. This means that any polynomial in k[x,y] can be written as a linear combination of monomials (terms with a single variable or a constant) with coefficients in k.

Next, we have the polynomial f(x,y) which is a member of k[x,y]. This means that we can write f(x,y) as a linear combination of monomials with coefficients in k. For example, f(x,y) could be written as f(x,y) = ax^2 + by^3 + cx + d, where a, b, c, and d are elements of k.

Now, let's focus on the specific term (x^3 - y^2). This is a polynomial in two variables, but it is not a member of k[x,y]. However, we can still use it as a factor in a polynomial in k[x,y], which is what we see in the statement f(x,y) = f_0(x) + f_1(x)y + (x^3 - y^2) g(x,y). This means that we are taking the polynomial f(x,y) and breaking it down into smaller parts, specifically f_0(x), f_1(x)y, and (x^3 - y^2) g(x,y).

The reason we can do this is because of the properties of polynomials and the distributive property. We know that if we have a polynomial in k[x,y], we can break it down into smaller parts by factoring out common terms. In this case, we are factoring out (x^3 - y^2) from f(x,y). This leaves us with f_0(x) + f_1(x)y, which are both polynomials in x with coefficients in k.

Lastly, we have the term g(x,y) which is also a polynomial in two variables. This term is multiplied by (x^3 - y^2), which means that it is a factor of f(x,y). This is similar to how we can write a polynomial like x^2 + 4x + 4 as (x + 2)^2.