Relatively Prime Quadratic Integers

In summary, a complex number can be written as the product of a unit and factors that have specific forms in the Gaussian integers Z(i). This allows for a unique factorization of complex numbers.
  • #1
stoolie77
7
0
Hello everybody. I found this example online and I was looking for some clarification.

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

Can someone shed some light on why this is so?

Many Thanks - Omar
 
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  • #2
(1+i)*(1-i)= 2.
 
  • #3
robert Ihnot said:
(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
 
  • #4
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.
 
  • #5
CRGreathouse said:
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:

[itex]32=2^5=(1-i)^5 (1+i)^5[/itex]

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

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

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

To me that seems like it would satisfy it!
 
  • #6
stoolie77 said:
is [itex]\alpha=(1-i)^5[/itex] , [itex]\beta=(1+i)^5[/itex] ? If so, then what would [itex]\epsilon[/itex] and [itex]\gamma[/itex] 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.
 
  • #7
stoolie77 said:
Can [itex]\alpha[/itex] be re-written so it is [itex](1-i)*(1-i)^4[/itex] thus making [itex]\epsilon[/itex] the first [itex](1-i)[/itex] term because it is a unit, and then [itex]\gamma[/itex] would be [itex](1-i)^2[/itex] ?

So CRGreathouse, my assumption above fails then?

And then you're saying that [itex]\epsilon[/itex] and [itex]\gamma[/itex] are both 1, and then you say that any factorization is of this form, not just this example, right?
 
  • #8
stoolie77 said:
So CRGreathouse, my assumption above fails then?

And then you're saying that [itex]\epsilon[/itex] and [itex]\gamma[/itex] 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.
 
  • #9
CRGreathouse said:
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?
 
  • #10
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
 

1. What are relatively prime quadratic integers?

Relatively prime quadratic integers are a pair of integers that share no common factors other than 1, and are both perfect squares. In other words, their greatest common divisor is 1 and they can be written as the square of two integers.

2. How are relatively prime quadratic integers different from regular prime numbers?

Unlike regular prime numbers which are indivisible by any number other than 1 and itself, relatively prime quadratic integers have to be perfect squares and their GCD must be 1. Additionally, there can be an infinite number of relatively prime quadratic integers, whereas there are only a finite number of prime numbers.

3. What are some examples of relatively prime quadratic integers?

Some examples of relatively prime quadratic integers include (3,4), (5,12), and (15,16). These pairs satisfy the criteria of being perfect squares and having a GCD of 1.

4. Can relatively prime quadratic integers be negative?

Yes, relatively prime quadratic integers can be negative. As long as the two numbers are both perfect squares and their GCD is 1, they can be considered relatively prime quadratic integers.

5. What is the significance of relatively prime quadratic integers?

Relatively prime quadratic integers have various applications in number theory, cryptography, and algebra. They also play a key role in the study of quadratic fields and have connections to complex numbers and elliptic curves.

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