Problem 1.11 asks us to solve some equations over the field of complex numbers. Specifically:
I feel like there's a way to use Euler's identity on the first three but I'm clueless on the last. Any thoughts?
Euler's identity and the related ideas from roots of unity in the book are definitely a winning path for the first three problems. Without giving it entirely away, on the fourth part I would recommend considering what factors might produce such a polynomial.
The fourth problem can be solved by substitution. We let $a = z^3$ and substitute in to get $$z^6 - z^3 - 2 = 0 \implies a^2 - a -2 = 0$$
This can be easily factored to get $$(a+1)(a-2) = 0$$ thus solving for $a = -1$ and $a = 2$. Now to solve in terms of $z$ have $z^3 = -1$ and $z^3 = 2$. The solutions to the first are $ \{ -1, e^\frac{\pi}{3}, e^\frac{-\pi}{3} \}$. To find the solutions to the second we use polar form, $$z^3 = 2 \implies r^3 e^{3\pi i} = 2$$ Since the imaginary part is 0 we can reason that for the argument $e^{3\theta i} = 1 $ only when $3\theta = 0$ or $3\theta = 2\pi $. The modulus is $\sqrt[3]{2}$. Thus the final solution set becomes $$ \{ -1, \sqrt[3]{2}, e^\frac{\pi}{3}, e^\frac{-\pi}{3}, \sqrt[3]{2}e^{\frac{2\pi i}{3}}, \sqrt[3]{2}e^{\frac{-2\pi i}{3}} \}$$.