Demonstrating the failure of the unique factorization into primes.

[burt]

Show that in the domain of integers of the form \(a+b\sqrt{-17}\) the factorization \(169=13\bullet13 =(4+3\sqrt{-17})(4-3\sqrt{-17})\) demonstrates that the unique factorization into primes fails in that domain.

I understand that this question is asking to show that it is not always possible to find a unique factorization into primes for complex numbers.

It seems to me that the norm of \((4+3\sqrt{-17})(4-3\sqrt{-17})\) is 169, which is \(13\bullet 13\).

A. How is this not a unique factorization into primes?
B. How can I demonstrate what the problem is asking?

 

[Dr.Peterson]

A. How is this not a unique factorization into primes?
B. How can I demonstrate what the problem is asking?

I believe they are suggesting that not only 13, but also [MATH](4+3\sqrt{-17})[/MATH] and [MATH](4-3\sqrt{-17})[/MATH] are primes in this domain. Can you show that?

 

[burt]

I believe they are suggesting that not only 13, but also [MATH](4+3\sqrt{-17})[/MATH] and [MATH](4-3\sqrt{-17})[/MATH] are primes in this domain. Can you show that?

If they are primes in this domain, does that not show that the prime factorization works?

 

[Dr.Peterson]

No, it shows that the prime decomposition is not unique!