1. Describing the Galois group adjoin root of unity

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!

2. Permutations of primitives

Each automorphism is determined entirely by where it sends one of the primitives. The fixed field is the same.

In your case just look to see the possibilities.
$\displaystyle e:\zeta \rightarrow \zeta$
$\displaystyle a:\zeta \rightarrow \zeta^3$
$\displaystyle b:\zeta \rightarrow \zeta^5$
$\displaystyle c:\zeta \rightarrow \zeta^7$

So you are correct in that it has order 4. It is pretty easy to see which one it is because there are only 2 groups of order 4. Klein 4 is characterized by the fact that every element has order 2. Cyclic 4 has some elements of order 4.

Check the order of these automorphisms (start with a) and see what you get.