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please could someone give me solution for this

Let $K$ be the field $\mathbb Q(\sqrt{−15})$, let $R = \mathcal{O}_K$ be the ring of integers of $K$. Let $\alpha= \frac{-1+\sqrt{-15}}{2}$ and consider the prime ideals $p = (2,α)$ and $q = (17,α + 6)$ of $R$.

Use the prime ideals p and q to construct an element of R that admits two distinct factorizations into irreducibles.

thank you

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Crossposted from MathOverflow. –  Zev Chonoles Feb 22 '13 at 0:39
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Since you are new, I want to give some advice about the site: To get the best possible answers, you should explain what your thoughts on the problem are so far. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people tend to be more willing to help you if you show that you've tried the problem yourself. If this is homework, please add the [homework] tag. –  Zev Chonoles Feb 22 '13 at 0:42
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1 Answer

Here is a hint (thanks to Gerry for the correction):

You should be able to show that (check!):

$(34)=(2)(17)=p\overline{p}q\overline{q}=pq\overline{p}\overline{q}$

How do you use this hint? Well, consider an easier case in $\mathbb{Z}[\sqrt{-5}]$, where we have

$(6)=(2)(3)=(1-\sqrt{-5})(1+\sqrt{-5})$

If you can figure out how this works, you'll be ready for your own question. If not, you should tell us what you don't understand about it.

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I think $p\overline{p}=(2)$. –  Gerry Myerson Feb 22 '13 at 3:55
    
Ahh, you are right. Thank you, makes the problem much easier. –  fixedp Feb 22 '13 at 3:57
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