The Fundamental Theorem of Algebra

Let the set $S=\{1-\frac{1}{2^{n}}\mid n\in\mathbb{N}\}$. Find $\inf S$ and $\sup S$, proving that the values found are indeed the infimum and supremum of $S$ respectively.

In the proof of 1.19, explain why $\mathrm{Gal}(L/\mathbb{R})$ has even order.

Let $z=x+yi\in\mathbb{C}$ where $x,y\in\mathbb{R}$. Prove that $\sqrt{z}\in\mathbb{C}$ rigorously using 1.18. That is, prove that there exists $w\in\mathbb{C}$ such that $w^{2}=z$.