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

Question

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?

Answer 1

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 $\mathbb{Q}(\zeta_5+\zeta_5^{-1})=\mathbb{Q}(\sqrt{5})$.