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Math Help - Relatively Prime Quadratic Integers

  1. #1
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    Relatively Prime Quadratic Integers

    Hello All,

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

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

    Could someone help explain how that statement is true?
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  2. #2
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    Can anyone explain how 32=\epsilon * \gamma^2 \beta ? This is just using the substitution described above.
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    Quote Originally Posted by Samson View Post
    Hello All,

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

    Assume that 32 = \alpha \beta for \alpha , \beta relatively prime quadratic integers in Q[i]. From here, it can be shown that \alpha = \epsilon * \gamma^2 for some unit \epsilon and some quadratic integer \gamma in 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.
    Attached Thumbnails Attached Thumbnails Relatively Prime Quadratic Integers-gaussiandivisorsof32.png  
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  4. #4
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    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.
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