• Sep 9th 2010, 02:49 PM
Samson
Hello All,

I was working on problems involving relatively prime quadratic integers. Here is the one that has me miffed.

Assume that $\displaystyle 32 = \alpha \beta$ for $\displaystyle \alpha , \beta$ relatively prime quadratic integers in $\displaystyle Q[i]$. From here, it can be shown that $\displaystyle \alpha = \epsilon * \gamma^2$ for some unit $\displaystyle \epsilon$ and some quadratic integer $\displaystyle \gamma$ in $\displaystyle Q[i]$.

Could someone help explain how that statement is true?
• Sep 17th 2010, 09:12 AM
Samson
Can anyone explain how $\displaystyle 32=\epsilon * \gamma^2 \beta$ ? This is just using the substitution described above.
• Sep 17th 2010, 10:43 AM
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Quote:

Originally Posted by Samson
Hello All,

I was working on problems involving relatively prime quadratic integers. Here is the one that has me miffed.

Assume that $\displaystyle 32 = \alpha \beta$ for $\displaystyle \alpha , \beta$ relatively prime quadratic integers in $\displaystyle Q[i]$. From here, it can be shown that $\displaystyle \alpha = \epsilon * \gamma^2$ for some unit $\displaystyle \epsilon$ and some quadratic integer $\displaystyle \gamma$ in $\displaystyle Q[i]$.

Could someone help explain how that statement is true?

Well there's non-elegant brute force,

Gaussian Divisors of an Integer - Wolfram Demonstrations Project

Haven't actually run the numbers. Of course there should be a better way, but for this few divisors, it wouldn't be so bad. And possibly a pattern would emerge allowing one to generalize and see the better solution.
• Sep 17th 2010, 04:59 PM
Samson
Well I downloaded the .nb file and opened it up in Mathematica, enabled the scale, but I'm still confused on how we determine 2 of them that are relatively prime and meet the other condition.

From what I see, we get values of 1,2,4,8, and 16. Am I missing something? None of these are relatively prime to one another.