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

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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–  Zev Chonoles Feb 22 '13 at 0:39

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