Prime Factorization Theorem and Number Systems

[e2m2a]

TL;DR

What does it mean that there are some number systems where the prime factorization theorem does not hold?

If you go to "The Abel Prize Interview 2016 with Andrew Wiles" on YouTube, there is a statement made by Andrew Wiles beginning at about 4:10 and ending about 4:54 where he mentions there are some new number systems possible where the fundamental theorem of arithmetic does not hold. I don't understand how this is possible. Can someone please explain what he meant by this and what number systems he is talking about?

 

[fresh_42]

Summary:: What does it mean that there are some number systems where the prime factorization theorem does not hold?

If you go to "The Abel Prize Interview 2016 with Andrew Wiles" on YouTube, there is a statement made by Andrew Wiles beginning at about 4:10 and ending about 4:54 where he mentions there are some new number systems possible where the fundamental theorem of arithmetic does not hold. I don't understand how this is possible. Can someone please explain what he meant by this and what number systems he is talking about?

Could you give us the link? At least to those who are willing to watch almost an hour of content only to guess what somebody else might have possibly meant.

$\mathbb{Z}[\sqrt{-5}]$ is a ring without prime factorization, because $2\cdot 3=(1+\sqrt{-5})\cdot(1-\sqrt{-5})$ are two different representations with irreducible factors.

 

[pbuk]

$\mathbb{Z}[\sqrt{-5}]$ is a ring without prime factorization, because $2\cdot 3=(1+\sqrt{-5})\cdot(1-\sqrt{-5})$ are two different representations with irreducible factors.

If you are not familiar with this notation then note that $\mathbb{Z}[\sqrt{-5}]$ is an example of a quadratic integer ring; specifically it is the 'number system' whose elements $w$ are defined by two (ordinary) integers $(a, b)$ such that $w = a + b \sqrt{-5}$.

As @fresh_42 states, in this number system the number $(6, 0)$ is an example of a number that has two distinct irreducible ('prime') factorizations: $(2, 0) (3, 0)$ and $(1, 1) (1, -1)$.

 

[e2m2a]

Could you give us the link? At least to those who are willing to watch almost an hour of content only to guess what somebody else might have possibly meant.

$\mathbb{Z}[\sqrt{-5}]$ is a ring without prime factorization, because $2\cdot 3=(1+\sqrt{-5})\cdot(1-\sqrt{-5})$ are two different representations with irreducible factors.

 

[cbarker1]

Another one is the a 2×2 matrix that has all the same integer entries. Will not have a fundamental theorem of arithmetic, I think the only unique factorization is a matrix of -1 or 1. There is a paper in the MAA Journal math monthly for more information about matrix number theory.

 

[martinbn]

Summary:: What does it mean that there are some number systems where the prime factorization theorem does not hold?

If you go to "The Abel Prize Interview 2016 with Andrew Wiles" on YouTube, there is a statement made by Andrew Wiles beginning at about 4:10 and ending about 4:54 where he mentions there are some new number systems possible where the fundamental theorem of arithmetic does not hold. I don't understand how this is possible. Can someone please explain what he meant by this and what number systems he is talking about?

The example given by @fresh_42 is exactly what Wiles is talking about. The rings of integers in number fields do not always have unique factorization. If they did Fermat's last thereom would have been much easier to prove.