Results 1 to 4 of 4

Thread: Elements in a finite field

  1. #1
    Newbie
    Joined
    May 2009
    Posts
    12

    Elements in a finite field

    Hello.

    I have to proof the following proposition:

    Let $\displaystyle K$ be a finite field. Then $\displaystyle K$ has $\displaystyle q=p^n$ elements, whereas $\displaystyle p$ is the characteristic of $\displaystyle K$ and $\displaystyle n=[K:\mathbb{F}_p]$.
    Furthermore I have to proof, that $\displaystyle x^q = x$ for all $\displaystyle x \in K$.

    Could anybody of you explain, how to proof the proposition?

    Bye,
    Lisa
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by lisa View Post
    Hello.

    I have to proof the following proposition:

    Let $\displaystyle K$ be a finite field. Then $\displaystyle K$ has $\displaystyle q=p^n$ elements, whereas $\displaystyle p$ is the characteristic of $\displaystyle K$ and $\displaystyle n=[K:\mathbb{F}_p]$.
    Furthermore I have to proof, that $\displaystyle x^q = x$ for all $\displaystyle x \in K$.

    Could anybody of you explain, how to proof the proposition?

    Bye,
    Lisa
    I'll assume p is a prime number. A finite field K of order $\displaystyle q=p^n$ is the splitting field of $\displaystyle x^q - x$ over $\displaystyle F_p$ (see here). Since a finite field is perfect, K is a separable extension of $\displaystyle F_p$. It follows that K is a Galois extension of a field $\displaystyle F_p$. By the fundamental theorem of Galois theory, $\displaystyle [K:F] = |Gal(K/F)|=n$, where $\displaystyle F=F_p$ and Gal(K/F) is a cyclic group of order n generated by a Frobenius automorphism.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    3
    Quote Originally Posted by lisa View Post
    Hello.

    I have to proof the following proposition:

    Let $\displaystyle K$ be a finite field. Then $\displaystyle K$ has $\displaystyle q=p^n$ elements, whereas $\displaystyle p$ is the characteristic of $\displaystyle K$ and $\displaystyle n=[K:\mathbb{F}_p]$.
    Furthermore I have to proof, that $\displaystyle x^q = x$ for all $\displaystyle x \in K$.

    Could anybody of you explain, how to proof the proposition?

    Bye,
    Lisa

    One of the easiest proofs is, perhaps, to note that any field is a vector space over any of its subfields, and then: a finite field $\displaystyle K$ of characteristic p is a v.s. over the prime field of char. p, $\displaystyle \mathbb{F}_p\cong \mathbb{Z}\slash p\mathbb{Z}$, obviously of finite dimension n, and choosing any basis for this v.s. a simple combinatoric reasoning gives that the number of elements in $\displaystyle K$ must be $\displaystyle p^n$

    Take the polynomial $\displaystyle f(x)=x^{p^n}-x\,\,\,over\,\,\mathbb{F}_p\left[x\right]$ . Now you can go with the argument from fields extensions and Galois theory that was given to you by Aliceinwonderland or else use a little group theory: the multiplicative group $\displaystyle K^{*}$ has order $\displaystyle p^n-1\,\Longrightarrow \omega^{p^n-1}=1\,\,\forall\,\,\omega\,\in\,K\,\Longrightarrow \,\omega^{p^n}=\omega$ , and we're done.

    Tonio
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    May 2009
    Posts
    12
    Thank you. Your explanations were very helpful for me.

    Best greets,
    Lisa
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Field with three elements
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Feb 1st 2013, 06:54 AM
  2. Splitting Field of a Polynomial over a Finite Field
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Apr 1st 2011, 03:45 PM
  3. finite fields of elements
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Apr 28th 2009, 01:55 PM
  4. The elements of a field
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 31st 2008, 03:10 PM
  5. [SOLVED] [SOLVED] matrix transformations with finite field elements
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: Mar 13th 2008, 01:15 PM

Search Tags


/mathhelpforum @mathhelpforum