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In my book - Algebraic Number Theory and Fermat's Last Theorem Ian Stewart, David Tall - there is the exercise:

  • Prove that the ring of integers of $\mathbb{Q}(\zeta_5)$ (the 5'th cyclotomic ring) is Euclidean.

I can prove that the integers of $\mathbb{Q}(\sqrt{-1})$ are Euclidean by a geometric argument but this doesn't work for the first problem since pentagons aren't in a lattice.

If anyone could give me a hint, thank you.

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Can you please tell us what your book is? – KCd Mar 22 '11 at 17:29
1 here's someones senior thesis on cyclotomic euclidean number fields – yoyo Mar 22 '11 at 17:40
No pentagons are involved; only parallelograms. – lhf Mar 22 '11 at 17:43
You need to consider the lattice generated by a basis of the ring of integers. You'll get a 4-dimensional parallelogram. Not easy to draw... – lhf Mar 22 '11 at 17:55
up vote 3 down vote accepted

To show that $\mathbb{Z}[\zeta_5]$ is norm-Euclidean, you need to show that for every $x\in\mathbb{Q}(\zeta_5)$ then there is some $y\in\mathbb{Z}[\zeta_5]$ with $\vert N(x-y)\vert < 1$. Here $N(\cdot)$ is the standard norm (product of conjugates in $\mathbb{Q}(\zeta_5)$.

As you mention, it is made easier in $\mathbb{Q}(\sqrt{-1})$ because the algebraic integers form a lattice. In more general numbers fields, you can still construct a lattice by considering the set of all real and complex embeddings. In this case, there are 4 complex embeddings $\mathbb{Q}(\zeta_5)\mapsto\mathbb{C}$ corresponding to $\zeta_5\mapsto\zeta_5^r$ ($r=1,2,3,4$). Write these as two embeddings, $\sigma_r(\zeta_5)=\zeta_5^r$ ($r=1,2$), together with their two complex conjugates $\bar\sigma_r$. Then, you can define a map $$ \begin{align} &f\colon\mathbb{Q}(\zeta_5)\to\mathbb{C}^2,\\ &f(x)=(\sigma_1(x),\sigma_2(x)). \end{align} $$ It is clear that $f(x+y)=f(x)+f(y)$ ($f$ is linear as a map of vector spaces over $\mathbb{Q}$). In fact, the image of $\mathbb{Z}[\zeta_5]$ is a lattice. Also, defining $\theta\colon\mathbb{C}^2\to\mathbb{R}$ by $\theta(x,y)=\vert xy\vert^2$, you can write the norm in $\mathbb{Q}(\zeta_5)$ as $N(x)=\theta(f(x))$. The problem reduces to trying to prove that for every $x\in\mathbb{C}^2$ there is an element $y$ of the lattice $\Lambda\equiv f(\mathbb{Z}[\zeta_5])$ satisfying $\theta(x-y) < 1$.

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HINT $\ \ $ It is norm Euclidean, i.e. the absolute value of the norm serves as a Euclidean function. For a nice survey see Lemmermeyer: The Euclidean algorithm in algebraic number fields.

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