1. Ring problem

Let F be a finite field if
f:F-->F given by f(x)=x^3 is a ring homomorphim THEN WITCH IS TRUE AND WHY
1)F=Z/3Z
2)F=Z/2Z or characteristic of F=3
3)F=Z/3Z or F=Z/2Z
4)characteristic of F=3

next question
let E be set of all rationals p such that 2< p^2< 3 then E is
1)compact in Q
2)not compact in Q
3)closed and bounded in Q
4))closed and unbounded in Q

2. Originally Posted by reflection_009
Let F be a finite field if
f:F-->F given by f(x)=x^3 is a ring homomorphim THEN WITCH IS TRUE AND WHY
1)F=Z/3Z
2)F=Z/2Z or characteristic of F=3
3)F=Z/3Z or F=Z/2Z
4)characteristic of F=3
For this problem use the fact that $(a+b)^p = a^p + b^p$ for $a,b\in F$ and $\text{char}(F) = p$.

let E be set of all rationals p such that 2< p^2< 3 then E is
1)compact in Q
2)not compact in Q
3)closed and bounded in Q
4))closed and unbounded in Q
$E = \{x \in \mathbb{Q}: 2 < x^2 < 3 \} = \{ x\in \mathbb{Q}: \sqrt{2} < x < \sqrt{3} \} \cup \{ x\in \mathbb{Q} : - \sqrt{3} < x < -\sqrt{2}\}$.

This is of course bounded.
Is it closed? No because we can find a sequence of point in $E$ converging to $\sqrt{3}$ which is not a rational number.