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Let $i$ be the primitive fourth root of unity and $E= \mathbb{Q}(i)=\{a_0 + a_1 i; a_0, a_1 \in \mathbb{Q}\}$ be a cyclotomic field of degree 2. Let $O_E$ be its ring of integers. I'm curious about the exact expression of $O_E$. I think this is well-studied in field theory, but I couldn't find any lemma about it in Dummit & Foote or online. Could anyone share some pointers for me to prove it?

Any help is appreciated! More details are below:

Let $\mathbb{Z}[i] = \{z_0 + z_1 i; z_0, z_1 \in \mathbb{Z}\}$. To show that $\mathbb{Z}[i] = O_E$, I could show that $\mathbb{Z}[i]$ is a subset of $O_E$, but I couldn't show every element in $O_E$ is an element in $\mathbb{Z}[i]$.

Sarah Li
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  • The standard arguments for this sort of thing involve computing the trace and the norm (which have to be integers for an element of $\mathcal{O}_E$); are you familiar with this? – Qiaochu Yuan Aug 15 '24 at 18:56
  • This has been asked several times here, see for example this duplicate. Your case is $d=-1$ in $\Bbb Q(\sqrt{d})$. – Dietrich Burde Aug 15 '24 at 18:59
  • @QiaochuYuan Umm, no I'm not familiar with this. I encountered the notion of $O_E$ when reading a paper related to quantum circuit characterization. There, the authors defined the Euclidean norm squared of an element $\alpha \in \mathbb{Q}(i)$ as $\langle \alpha,\alpha \rangle=1/2Tr_{E/\mathbb{Q}}(\alpha \alpha^{*}) = 1/2\sum_{j \in [2d]}\sigma_j(\alpha)$. Here $2d=2$ and $\sigma_j$ is an automorphism from $\mathbb{Q}(i)$ to $\mathbb{Q}(i)$. – Sarah Li Aug 15 '24 at 19:05
  • @DietrichBurde Thank you so much for pointing me to this post. I'll read up on this. If this solves my confusion, I'll close the question. – Sarah Li Aug 15 '24 at 19:09
  • I found a perhaps better duplicate, giving a very elementary and clear proof for ℚ(). Let me know if you like it. It uses the Gauss lemma. – Dietrich Burde Aug 15 '24 at 19:26

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