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

In summary, the conversation is about a specific theorem in abstract algebra and the proof of it. The conversation includes a link to the proof and a discussion about the meaning of certain variables and their relations to each other. The summary concludes by restating the main point of the conversation, which is that the theorem proves that a certain variable, a_1, divides another variable, b_1 b_2 ... b_n.
  • #1
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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 \(\displaystyle a = a_1 a_2 \ ... \ ... \ a_m = b_1 b_2 \ ... \ ... \ b_n\) where each \(\displaystyle a_i\) and \(\displaystyle b_i\) is irreducible, then \(\displaystyle a_1 \mid b_1 b_2 \ ... \ ... \ b_n\) ... ... "
Can someone please explain exactly and in detail why/how \(\displaystyle a_1 \mid b_1 b_2 \ ... \ ... \ b_n\) ... ... Peter
 
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  • #2
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 $y$ ?
 
  • #3
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:

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

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

Peter
 
  • #4
Correct, and do you understand that therefore $a_1|b_1 \cdots b_n$ ?
 

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

What is a Principal Ideal Domain (PID)?

A Principal Ideal Domain (PID) is a type of mathematical structure in abstract algebra that satisfies certain properties. Specifically, in a PID, every ideal (a subset of the ring) can be generated by a single element. This means that every element in the ring can be expressed as a multiple of this single generating element.

What is a Unique Factorization Domain (UFD)?

A Unique Factorization Domain (UFD) is also a type of mathematical structure in abstract algebra, and it is a special type of PID. In a UFD, every nonzero, nonunit element can be written as a unique product of irreducible elements (elements that cannot be further factored). This is known as the fundamental theorem of arithmetic.

What is Bland's AA Theorem (Theorem 7.2.20)?

Bland's AA Theorem is a result in abstract algebra that relates principal ideal domains and unique factorization domains. It states that every PID is also a UFD, meaning that every principal ideal domain satisfies the fundamental theorem of arithmetic. This is a very important result in abstract algebra, as it connects two important mathematical structures.

What are some examples of Principal Ideal Domains and Unique Factorization Domains?

Some examples of principal ideal domains include the integers (ℤ), the Gaussian integers (ℤ[i]), and the polynomial ring ℤ[x]. Examples of unique factorization domains include the integers (ℤ), the Gaussian integers (ℤ[i]), and the polynomial ring F[x] over any field F.

How are Principal Ideal Domains and Unique Factorization Domains used in mathematics?

Principal Ideal Domains and Unique Factorization Domains have many applications in mathematics, especially in number theory and algebraic geometry. They are used to study and prove properties of numbers and polynomials, and they have important connections to other mathematical structures such as fields and vector spaces. They are also used in cryptography, coding theory, and other areas of applied mathematics.

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