Relatively Prime Quadratic Integers

**Samson** · December 8th 2010, 10:44 AM

Hello Math Help Forum,

I saw the following problem on another forum but the responses were all over the place and I was hoping if someone could give some clarity. The problem states:

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

Can someone explain how this is shown?

**Bruno J.** · December 29th 2010, 01:04 PM

Well, $32 = 2^5 = (1+i)^5(1-i)^5$ . And $1+i,1-i$ are irreducible. But $(1+i)=i(1-i)$ . Hence $32 = i(1-i)^{10}$ is the unique expression of $32$ in $\mathbb{Z}[i]$ as a product of irreducibles. Can you finish from there?

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