MHB Why Does \( a_1 \mid b_1 b_2 \cdots b_n \) in Theorem 7.2.20?

Math Amateur
Gold Member
MHB
Messages
3,920
Reaction score
48
I am reading The Basics of Abstract Algebra by Paul E. Bland ...

I am focused on Section 7.2 Euclidean, Principal Ideal, Unique Factorization Domains ... ...

I need help with the proof of Theorem 7.2.20 ... ... Theorem 7.2.20 and its proof reads as follows:https://www.physicsforums.com/attachments/8280
View attachment 8281
In the last paragraph of the above proof by Bland we read the following:

" ... ... If $$a = a_1 a_2 \ ... \ ... \ a_m = b_1 b_2 \ ... \ ... \ b_n$$ where each $$a_i$$ and $$b_i$$ is irreducible, then $$a_1 \mid b_1 b_2 \ ... \ ... \ b_n$$ ... ... "
Can someone please explain exactly and in detail why/how $$a_1 \mid b_1 b_2 \ ... \ ... \ b_n$$ ... ... Peter
 
Physics news on Phys.org
Notate $x=b_1 b_2 \cdots b_n$
What does it mean if $a_1$ divides $x$ ?
It means that there is a $y \in D$ such that $a_1y=x$
Can you tell now what is $y$ ?
 
steenis said:
Notate $x=b_1 b_2 \cdots b_n$
What does it mean if $a_1$ divides $x$ ?
It means that there is a $y \in D$ such that $a_1y=x$
Can you tell now what is $y$ ?
Thanks for the help Steenis ...

Basically you have pointed out that:

$$a_1 ( a_2 a_3 \ ... \ ... \ a_m ) = b_1 b_2 \ ... \ ... \ b_n $$

$$\Longrightarrow a_1 \mid b_1 b_2 \ ... \ ... \ b_n$$The above implies that in what you have written we have $$y = a_2 a_3 \ ... \ ... \ a_m$$ ... ...Is that correct?

Peter
 
Correct, and do you understand that therefore $a_1|b_1 \cdots b_n$ ?
 
I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
Back
Top