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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:View attachment 5680In 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 $$<f_1, h> \ = \ <f_1, \ ... \ ... \ , f_s>$$ ... ...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
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:View attachment 5680In 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 $$<f_1, h> \ = \ <f_1, \ ... \ ... \ , f_s>$$ ... ...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