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## Homework Statement

I am reading the undergraduate introduction to algebraic geometry entitled "Ideals, Varieties and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra (Third Edition) by David Cox, John Little and Donal O'Shea ... ...

I am currently focused on Chapter 1, Section 5: Polynomials of One Variable ... ... and need help with the proof of Proposition 8, part 3 ...

Proposition 8 of Chapter 1 (including Definition 7 which is relevant) reads as follows:

In the above text from Cox et al we read the following:

" ... ... To prove part (iii), let ##h \ = \ GCD(f_2, \ ... \ ... \ , f_s)##. We leave it as an exercise to show that

##<f_1, h> \ = \ <f_1, \ ... \ ... \ , f_s>##

... ... "

I need help to show that ##<f_1, h> \ = \ <f_1, \ ... \ ... \ , f_s>## ... ...

## Homework Equations

These are all introduced in context, in 3 below ... ...

## The Attempt at a Solution

Work so far ...

We need to show that ##<f_1, h> \ \subset \ <f_1, \ ... \ ... \ , f_s>## ... and also that

##<f_1, \ ... \ ... \ , f_s> \ \subset \ <f_1, h>##

So to show ##<f_1, h> \ \subset \ <f_1, \ ... \ ... \ , f_s>## we start with

Let ##l \in <f_1, h>## ...

Then, by definition of ##<f_1, h>##, we have that ##l = f_1 t_1 + h t_2 where t_1, t_2 \in k[x]## ...

Now we have that ##h = GCD(f_2, \ ... \ ... \ , f_s)## ... BUT ... how do we use this in the proof?

Note that we also have

##(1) \ h = GCD(f_2, \ ... \ ... \ , f_s) \ \Longrightarrow \ h \text{ divides } f_2, \ ... \ ... \ , f_s##

## \ \Longrightarrow \ h = f_2 u_2, h \ = \ f_3 u_3, \ ... \ ... \ , h \ = \ f_s u_s##

for some ##u_2, \ ... \ ... \ , u_s \in k[x]## ...

(2) ##k[x]## is a PID so that:

##<f_1, h > \ = \ <v>## for some polynomial ##v \in k[x]## ... ...

... but, how do we use (1) and (2) in the required proof ...

Can someone please help me to complete the proof of ##<f_1, h> \ = \ <f_1, \ ... \ ... \ , f_s>## ...

Help will be appreciated ...

Peter