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Math Help - Relatively prime quadratic integers

  1. #1
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    Relatively prime quadratic integers

    I've been stuck on this problem for a while now.

    Suppose 32 = \alpha\beta for \alpha,\beta relatively prime quadratic integers in \mathbb{Q}[i]. Show that \alpha=\epsilon\gamma^{2} for some unit \epsilon and some quadratic integer \gamma in \mathbb{Q}[i].

    Any pointers on how to start?

    Thanks.
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  2. #2
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    Quote Originally Posted by Pn0yS0ld13r View Post
    I've been stuck on this problem for a while now.

    Suppose 32 = \alpha\beta for \alpha,\beta relatively prime quadratic integers in \mathbb{Q}[i]. Show that \alpha=\epsilon\gamma^{2} for some unit \epsilon and some quadratic integer \gamma in \mathbb{Q}[i].

    Any pointers on how to start?

    Thanks.
    \alpha \beta=32=2^5=(i(1-i)^2)^5=i(1-i)^{10}. we know that i is a unit and 1-i is irreducible in \mathbb{Z}[i], the ring of integers of \mathbb{Q}[i]. now use this fact that \mathbb{Z}[i] is a UFD to finish the proof.
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  3. #3
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    I had a different proof (NonCommAlg beat me ) that did not involve finding factorizations.
    Note that N(32) = 2^{10}. Therefore, if \alpha (a non-unit) is decomposed into irreducibles it must mean that norm of each irreducible is a divisor of 2^{10}. Since 2 is not a prime and N(2) = 4 it means all irreducibles must have norm 2. Note that 1+i is the only irreducible of norm 2 up to associates. This immediately forces \beta to be a unit for otherwise it would mean \alpha,\beta are not relatively prime. Thus, \alpha = u(1+i)^{10} for some unit u.
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  4. #4
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    Thank you NonCommAlg and ThePerfectHacker! I got it now.
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