Relatively Prime Quadratic Integers

[stoolie77]

Hello everybody. I found this example online and I was looking for some clarification.

Assume $32 = \alpha\beta$ for $\alpha,\beta$ relatively prime quadratic integers in $\mathbb{Q}$. It can be shown that $\alpha = \epsilon \gamma^2$ for some unit $\epsilon$ and some quadratic $\gamma$ in $\mathbb{Q}$.

Can someone shed some light on why this is so?

Many Thanks - Omar

 

[robert Ihnot]

(1+i)*(1-i)= 2.

 

[stoolie77]

(1+i)*(1-i)= 2.

Hello Robert, could you expand on this? I don't see how it directly relates to my example of 32 considering you used values that net a 2. Could you use some of the variables as well (some sort of General notation)? I'm just lost in connecting what you're saying to what I'm trying to figure out.

Many Thanks - Omar

 

[CRGreathouse]

I'm almost hesitant to point out that 2^5 = 32, and so any factor of 32 is of the form unit * (1 - i)^a * (1 + i)^b.

 

[stoolie77]

I'm almost hesitant to point out that 2^5 = 32, and so any factor of 32 is of the form unit * (1 - i)^a * (1 + i)^b.

Thank you CR Greathouse! I'm still confused on how to relate this to the variables though and what you have written above. I think this is what you have written above:

$32=2^5=(1-i)^5 (1+i)^5$

So when I try to relate that to something of the form $32=\alpha\beta$ where $\alpha=\epsilon \gamma^2$,

is $\alpha=(1-i)^5$ , $\beta=(1+i)^5$ ? If so, then what would $\epsilon$ and $\gamma$ be?

Can $\alpha$ be re-written so it is $(1-i)*(1-i)^4$ thus making $\epsilon$ the first $(1-i)$ term because it is a unit, and then $\gamma$ would be $(1-i)^2$ ?

To me that seems like it would satisfy it!

 

[CRGreathouse]

is $\alpha=(1-i)^5$ , $\beta=(1+i)^5$ ? If so, then what would $\epsilon$ and $\gamma$ be?

1 and 1 would work. But what you need to show is that any factorization is of this form, not just that there is some factorization of this form. You now have all the tools you need to prove that.

 

[stoolie77]

Can $\alpha$ be re-written so it is $(1-i)*(1-i)^4$ thus making $\epsilon$ the first $(1-i)$ term because it is a unit, and then $\gamma$ would be $(1-i)^2$ ?

So CRGreathouse, my assumption above fails then?

And then you're saying that $\epsilon$ and $\gamma$ are both 1, and then you say that any factorization is of this form, not just this example, right?

 

[Brimley]

So CRGreathouse, my assumption above fails then?

And then you're saying that $\epsilon$ and $\gamma$ are both 1, and then you say that any factorization is of this form, not just this example, right?

I believe this fails but perhaps CRGreathouse could explain why better than I could.

 

[ramsey2879]

I'm almost hesitant to point out that 2^5 = 32, and so any factor of 32 is of the form unit * (1 - i)^a * (1 + i)^b.

So then is it true that the prime factorization of a complex number is unique? That is is it true that if a,b,c,d are each unique primes in the complex number system, then ab <> cd?

 

[lpetrich]

It can be shown that Gaussian integers Z(i) (integer + i*integer) can be uniquely factored as:

(unit)*(factors f with form Re(f) >= |Im(f)| and Re(f) > 0)

where (unit) is any of 1, -1, i, and -i.

I've found a page with some more detail on that subject: Integral Domains, Gaussian Integer, Unique Factorization