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Thread: 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 $\displaystyle 0 \neq a \in K,$ then $\displaystyle (a^{-1})^p=(a^p)^{-1}=a^{-1}.$ so $\displaystyle a^{-1} \in K.$ also if $\displaystyle a,b \in K,$ then $\displaystyle (ab)^p=a^pb^p=ab.$ so $\displaystyle ab \in K.$


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

    let $\displaystyle [F:\mathbb{F}_p]=n.$ then $\displaystyle |F|=p^n$ because $\displaystyle |\mathbb{F}_p|=p.$
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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: $\displaystyle (a+b)^p = a^p + b^p \text{ and }(ab)^p = a^pb^p$.

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

    EDIT: Too slow.
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