Finite extension fields of odd degree over $F_{2}$

Question

I'm actually getting stuck on an earlier part of this problem, but once I have it I think the rest is clear. First (part (a)), it's obvious that for positive, odd $k, 2$ is a unit in $\mathbb{Z}/k \mathbb{Z}$. Now I'm trying to prove part (b), that there exists a positive integer $\ell$ such that $2^{\ell} = \overline{1}$ in $\mathbb{Z}/k \mathbb{Z}$. This is obvious with small examples, but I'm missing how to show it in general. Please, just a hint to start with, and I'll add a comment if I'm giving up.

Building on this question, I need to show (part (c)) that there exists a finite extension field $F$ of the field $\mathbb{F}_{2} = \mathbb{Z}/2\mathbb{Z}$ such that all $k$'th roots of unity exist in $F$. Unless I'm mistaken, this should be $\mathbb{F}_{2}[x]/(x^{k} - 1)$. And for a particular $k$ (part (d)), the smallest such extension should be $\mathbb{Q}(\zeta_{k})$.

Answer 1

This result holds in an abstract setting. Namely every element of a finite group has finite order. Hint:

If $G$ is a finite group and $x \in G$ then the powers $x^0, x^1, x^2, x^3, \dots$ cannot all be distinct.

What is the relevant group that we can use to show that $2^k = 1$ in $\mathbf{Z}/k\mathbf{Z}$? Answer:

The group of units: $\{a \in \mathbf{Z}/k\mathbf{Z} : \gcd(a,k) = 1\}$ with multiplication as the group operation.