Determine Units of a Ring $\mathbb{Z}[\alpha]$

I am trying to determine the units of $Z[\alpha]$ where $\alpha$ satisfies the monic polynomial $\alpha^4+\alpha^3+\alpha^2+\alpha+1$. I found $Z[\alpha] := \lbrace a+b\alpha+c\alpha^2+d\alpha^3\;|\; a,b,c,d \in \mathbb{Z}\rbrace$. I need to find elements that satisfy $st=1$ where $s,t \in \mathbb{Z}[\alpha]$ I found the multiplicative inverse of $a+...+d\alpha^3$ to be a very nasty diophantine equation with 4 variables, so I am not sure in which way am I suppose to approach this problem?

• Note that $(x-1)(x^4+x^3+x^2+1)=x^5-1$. – Adam Hughes Jul 1 '14 at 4:34
• First of all it tells you you are looking at the field $\mathbb{Q}(\zeta_5)$ which is handy info, since you can use Dirichlet's unit theorem to get some info about the structure of the units. – Adam Hughes Jul 1 '14 at 4:38
• Mind you there's no general, simple description for units, even in a cyclotomic integer ring; as you note: it's messy. – Adam Hughes Jul 1 '14 at 4:43
• It’s late, and I have to turn in. If you haven’t gotten anywhere on this by tomorrow, and if no-one else gives you a hand, I can help a little. But my question to you now is whether you have identified the (unique) quadratic subfield, and noted that it’s real. Its fundamental unit is well known, and generates a subgroup of finite index in the unit group of your ring. – Lubin Jul 1 '14 at 5:28
• The moral of the whole story is that this is deep mathematics indeed, and there would have been no hope whatever of reaching a good result by the very elementary methods you were starting out with. – Lubin Jul 1 '14 at 13:02

By Dirichlet's unit theorem, applied to the ring of integers $\mathcal{O}_K=\mathbb{Z}[\zeta_5]$ for the number field $\mathbb{Q}(\zeta_5)$, the unit group $\mathcal{O}_K^*$ is isomorphic to $\mathbb{Z}^{\frac{5-3}{2}}\times \mu_K\simeq \mathbb{Z}\times \langle -\zeta_5\rangle \simeq \mathbb{Z}\times \mathbb{Z}/10$. A fundamental unit is given by $\frac{1-\zeta_5^2}{1-\zeta_5}=1+\zeta_5$. Note that $\mathbb{Q}(\zeta_5+\zeta_5^{-1})=\mathbb{Q}(\sqrt{5})$.