/sci/ - Science & Math

I'm giving up on my algebra.

Problems like these take a hell lot of time to solve + solutions are ridiculously long.

Please look at these and say: are they really this hard or maybe I'm missing something? Please answer, I need to know:

1. For which [math]d \in \{2, 3, \ldots 10\}[/math] ring [math]\mathbb{Z}+\mathbb{Z}[\sqrt{d}][/math] is integrally closed?
2. Prove that the quotient ring of [math]\frac{\mathbb{C}[x, y, z]}{\left< x^3 + y^3 +z^3 - 1 \right>}[/math] is isomorphic to the field of rational functions of two variables.
3. How many solutions does [math]x^2+y^2=1[/math] have in [math]\mathrm{F}_q[/math] (q is prime)?