Prime and Irreducible Elements in Principal Ideal Domains .... Bland - AA - Theorem 7.2.14 .... ....

[Math Amateur]

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 7.2.14 ... ... Theorem 7.2.14 and its proof reads as follows:
View attachment 8275In the above proof by Bland we read the following:

"... ... But then \(\displaystyle a = apx' + aby' \), so \(\displaystyle p|apx'\) and \(\displaystyle p|aby'\). ... ... "Can someone please explain exactly how/why \(\displaystyle p|aby'\)... ... ?

Peter

 

[castor28]

Hi Peter,

That comes from the assumption that $p\mid ab$ (on line 3 of the proof).

 

[Math Amateur]

Hi Peter,

That comes from the assumption that $p\mid ab$ (on line 3 of the proof).

Thanks castor28 ...

Peter

 

[Olinguito]

This is a very important theorem in the study of rings. There are some definitions not mentioned in the OP and it is worth filling them in.

Let $D$ be an integral domain. An element $x\in D$, $x\ne0,e$, is said to be prime iff for any elements $a,b\in D$, if $x\mid ab$ then either $x\mid a$ or $x\mid b$. It is irreducible iff whenever $x=ab$ then either $a$ or $b$ is a unit (divisor of the multiplicative identity $e$). It easily follows from these definitions that in any domain $D$ every prime element is irreducible. The theorem says that when $D$ is a PID, the converse is also true.

This may not be how we are used to thinking of primes when dealing with the integers $\mathbb Z$. When we say that an integer $p\ne0,1$ is prime, we tend to think of $p$ as having no divisors other than $\pm1,$ and $\pm p$. Furthermore, $\mathbb Z$ is an ordered ring (i.e. has positive and negative elements) and so saying that $p\in\mathbb Z$ is prime usually means $p>1$ and has no factors other than $1$ and $p$. Strictly speaking, this is only thinking of $p$ as an irreducible rather than prime. However, as $\mathbb Z$ is a PID, the two definitions coincide and so the distinction is immaterial.

The distinction in the two definitions becomes important in the study of ideals and UFDs (unique-factorization domains) in algebraic-number theory. This branch of mathematics was initially developed to tackle Fermat’s last theorem: that the equation $x^n+y^n=z^n$ has no nonzero integer solutions in $x,y,z$ if $n$ is an integer greater than 2. The theory culminated in Kummer’s proof that the result holds if $n$ is a regular prime – which was the furthest the theory could go. The general proof continued to elude mathematicians until 1995, when Andrew Wiles succeeded in proving a special case of the Taniyama–Shimura conjecture relating elliptic curves and modular forms.

 

[Math Amateur]

This is a very important theorem in the study of rings. There are some definitions not mentioned in the OP and it is worth filling them in.

Let $D$ be an integral domain. An element $x\in D$, $x\ne0,e$, is said to be prime iff for any elements $a,b\in D$, if $x\mid ab$ then either $x\mid a$ or $x\mid b$. It is irreducible iff whenever $x=ab$ then either $a$ or $b$ is a unit (divisor of the multiplicative identity $e$). It easily follows from these definitions that in any domain $D$ every prime element is irreducible. The theorem says that when $D$ is a PID, the converse is also true.

This may not be how we are used to thinking of primes when dealing with the integers $\mathbb Z$. When we say that an integer $p\ne0,1$ is prime, we tend to think of $p$ as having no divisors other than $\pm1,$ and $\pm p$. Furthermore, $\mathbb Z$ is an ordered ring (i.e. has positive and negative elements) and so saying that $p\in\mathbb Z$ is prime usually means $p>1$ and has no factors other than $1$ and $p$. Strictly speaking, this is only thinking of $p$ as an irreducible rather than prime. However, as $\mathbb Z$ is a PID, the two definitions coincide and so the distinction is immaterial.

The distinction in the two definitions becomes important in the study of ideals and UFDs (unique-factorization domains) in algebraic-number theory. This branch of mathematics was initially developed to tackle Fermat’s last theorem: that the equation $x^n+y^n=z^n$ has no nonzero integer solutions in $x,y,z$ if $n$ is an integer greater than 2. The theory culminated in Kummer’s proof that the result holds if $n$ is a regular prime – which was the furthest the theory could go. The general proof continued to elude mathematicians until 1995, when Andrew Wiles succeeded in proving a special case of the Taniyama–Shimura conjecture relating elliptic curves and modular forms.

Thanks Olinguito ...

The above is a most interesting and helpful post ...

Thanks again,

Peter