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I would like to show that $\mathbb{Z}[\phi]$ is a UFD, where $\phi^2+\phi+1=0$. I started with a division with remainder to determine the irreducibles, but I could not finish that, and I do not know how to approach this problem. Could you provide me with some resources or show how I can solve this? Thanks in advance.

Mrtired
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  • This ring is the ring of integers of $\mathbb{Q}(\sqrt{5})$. Since $\mathbb{Q}(\sqrt{5})$ has class number $1$, its ring of integers is a PID and a UFD. – Dietrich Burde Mar 12 '18 at 12:48
  • I don't know the topic you are referring to. I think I am expected to solve this by defining a norm. I am sort of clueless. – Mrtired Mar 12 '18 at 13:00
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    The standard proof is to show that the ring is a PID. Then it is a UFD of course. But this is just class number $1$ and follows immediately from the fact that $B_K<2$ (Minkowski bound). Alternatively show that every ideal is principal. – Dietrich Burde Mar 12 '18 at 13:13
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    I believe this is the ring of Eisienstein integers where $\phi$ is a primitive cube root of unity. Show this ring is a UFD by showing it is Euclidean: For $\alpha, \beta \in \mathbb Z[\phi]$, we can write $\alpha/\beta = p + q\phi, p,q \in \mathbb Q$. Now round $p$ and $q$ to the nearest integers $m,n$ and show that $N[\alpha-(m+n\phi) \beta < N(\beta)$ and thus that $\mathbb Z[\phi]$ has a division algorithm. – John Brevik Mar 12 '18 at 14:06

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