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Math Help - 2 Questions: Fields and Subfields

  1. #1
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    2 Questions: Fields and Subfields

    1.) F is a field of prime characteristic p. Prove that K = {x in F for all x^p=x} is a subfield of F.

    For the subfield test, I showed that a-b is in K whenever a and b are in K, but I'm getting stuck when I try to show that ab inverse is in K.

    2.) Show that any finite field has order p^n, where p is prime.

    I have that Char F = p and the order of a divides p, and I know that the order of a is either 1 or p. I also know that p divides the order of the field, so F must contain an element of order p. I'm just having trouble connecting that to proving any finite field has order p^n.

    Any help would be greatly appreciated. Thanks.
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  2. #2
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    Quote Originally Posted by GB89 View Post
    1.) F is a field of prime characteristic p. Prove that K = {x in F for all x^p=x} is a subfield of F.

    For the subfield test, I showed that a-b is in K whenever a and b are in K, but I'm getting stuck when I try to show that ab inverse is in K.
    continuing what you've already done: 0 and 1 are in K. also if 0 \neq a \in K, then (a^{-1})^p=(a^p)^{-1}=a^{-1}. so a^{-1} \in K. also if a,b \in K, then (ab)^p=a^pb^p=ab. so ab \in K.


    2.) Show that any finite field has order p^n, where p is prime.

    I have that Char F = p and the order of a divides p, and I know that the order of a is either 1 or p. I also know that p divides the order of the field, so F must contain an element of order p. I'm just having trouble connecting that to proving any finite field has order p^n.

    Any help would be greatly appreciated. Thanks.
    f: \mathbb{Z} \longrightarrow F defined by f(n)=n1_F is a ring homomorphism and \ker f = p \mathbb{Z}. thus f contains a field \mathbb{F}_p \simeq \frac{\mathbb{Z}}{p\mathbb{Z}}. hence F can be considered as a finite dimensional vector space over \mathbb{F}_p.

    let [F:\mathbb{F}_p]=n. then |F|=p^n because |\mathbb{F}_p|=p.
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  3. #3
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    Quote Originally Posted by GB89 View Post
    1.) F is a field of prime characteristic p. Prove that K = {x in F for all x^p=x} is a subfield of F.
    Hint: (a+b)^p = a^p + b^p \text{ and }(ab)^p = a^pb^p.

    2.) Show that any finite field has order p^n, where p is prime.
    Define \phi:\mathbb{Z}\to F by \phi(n) = n\cdot 1 where 1 is multicative identity element of F.
    Prove that \phi is a ring homomorphism and \ker(\phi) = p\mathbb{Z} since \text{char}(F) = p.
    It means that \mathbb{Z}_p\simeq \phi[ \mathbb{Z}]\subseteq F.
    Therefore, F contains a subfield of order p.
    Since F is a vector space over this smaller field forces |F| = p^n for some n.

    EDIT: Too slow.
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