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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
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