Taking $\displaystyle \zeta$ to be the first primitive root of unity ($\displaystyle \zeta=e^{i\pi/4} $), I am trying to describe $\displaystyle \Gamma[\mathbb{Q}(\zeta):\mathbb{Q}]$ and I'm not sure how to be completely sure that I've categorized the group. I mean, I am unsure how to tell what the structure of the group is: ie. $\displaystyle \mathbb{Z}_4$ or $\displaystyle \mathbb{Z}_2\oplus\mathbb{Z}_2$

I am pretty sure that it has order 4, but I may be wrong... Any direction would be greatly appreciated! Thanks!