Obtain an unit in the integral group ring $\mathbb{Z}[\mathbb{Z}/5\mathbb{Z}]$ Obtain a nontrivial unit of the integral group ring $\mathbb{Z}[\mathbb{Z}/5\mathbb{Z}]$. By nontrivial I mean not a group element.
Since brute force doesn't seem possible here, I tried to guess what should be a unit, other than the group elements. But I have failed in this attempt. I would appreciate any kind of help.
Thank you!

Hint: the group ring is isomorphic to $\mathbb{Z}[\zeta_{5}]$, where $\zeta_5$ is a 5th root of unity, which is a ring of integer $\mathbb{Q}(\zeta_{5})$. In fact, we can even compute the unit group - see Exercise 4 in Chapter 1.7 of Neukirch, Algebraic number theory. 
Note that $1+\zeta_5$ is a unit of infinite order. 

Edit: As Alex Wertheim said, $\mathbb{Z}[\mathbb{Z}/5\mathbb{Z}]$ is NOT isomorphic to $\mathbb{Z}[\zeta_5]$, but isomorphic to $\mathbb{Z}\times \mathbb{Z}[\zeta_5]$. 
