Gaussian integers

[Kcashew]

Here is the problem I am faced with:

In class we discussed how a prime p could be factored using complex Gaussian integers as (a + bi)(a − bi) where a and b are integers. If such a factorization exists, why must the integers a and b have no common (non-trivial) factors?


Here is my solution:

The resulting integer would no longer be a prime. For example, if a and b were equal to one, the product would equal 0.


Am I correct in my assertion? How would I further correct my statement?

 

[HallsofIvy]

If a and b were 1 the product would be (1+ i)(1- i)= 2, NOT 0! I don't think you understand what the problem is asking.

 

[Kcashew]

I think I understand now.

No matter what they are, if a and b are common factors, the product will always be 0, and never any other number.

Am I correct in this statement?

 

[lex]

[MATH]p[/MATH] prime and [MATH]p=\left(a+bi\right)\left(a-bi\right)=a^2+b^2[/MATH], where [MATH]a, b[/MATH] are integers
[MATH]p=a^2+b^2[/MATH] where [MATH]a, b[/MATH] are integers

So the question is, why might this imply that [MATH]a[/MATH] and [MATH]b[/MATH] cannot have a (non-trivial) common factor?