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Math Help - Ring problem

  1. #1
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    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

    please also give reasons thanks
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  2. #2
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    Quote Originally Posted by reflection_009 View Post
    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.
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