Results 1 to 2 of 2

Math Help - fields and elements

  1. #1
    Member
    Joined
    Jan 2010
    Posts
    104

    fields and elements

    Let K be a finite field with k elements. Let K*={x1,x2,...,xk-1}, the non-zero elements of K.
    Show that product (x belonging to K*) x=x1x2...xk-1=-1.

    Hint: Try to arrange, in the product, the elements in pairs a,b with ab=1, i.e. b=a^(-1) to the extent possible. Of course, a can not equal a^(-1) Which elements aren't paired?

    I'm highly confused by this. I get it in the concrete example, i.e. in Z11, we get an arrangement of this form (2*6)(3*4)(5*9)(7*8) (1)(10), but I'm having trouble with it in the abstract.

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by twittytwitter View Post
    Let K be a finite field with k elements. Let K*={x1,x2,...,xk-1}, the non-zero elements of K.
    Show that product (x belonging to K*) x=x1x2...xk-1=-1.

    Hint: Try to arrange, in the product, the elements in pairs a,b with ab=1, i.e. b=a^(-1) to the extent possible. Of course, a can not equal a^(-1) Which elements aren't paired?

    I'm highly confused by this. I get it in the concrete example, i.e. in Z11, we get an arrangement of this form (2*6)(3*4)(5*9)(7*8) (1)(10), but I'm having trouble with it in the abstract.

    Thanks.
    DISCLAIMER: You may want to wait for verification by another member. I am NOT too experienced in field theory.


    For every k\in K-\{1,-1\} there exists some k' such that kk'=1. Thus, \prod_{k\in K}k=\prod_{k\in  K-\{1,-1\}}k\cdot 1\cdot -1=(k_1k'_1)(k_2k'_2)\cdots =-1\cdot1\cdots=-1
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Elements of Extension Fields
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 19th 2011, 05:38 PM
  2. Replies: 1
    Last Post: September 23rd 2010, 10:59 AM
  3. vector fields - graphing gradient fields
    Posted in the Calculus Forum
    Replies: 0
    Last Post: March 20th 2010, 05:53 PM
  4. Basisrepresentation of elements in fields
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 4th 2010, 11:43 AM
  5. finite fields of elements
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 28th 2009, 01:55 PM

Search Tags


/mathhelpforum @mathhelpforum