Results 1 to 3 of 3

Math Help - Proving a field is not algebraically closed

  1. #1
    Newbie
    Joined
    Apr 2009
    Posts
    4

    Proving a field is not algebraically closed

    Hi,

    My problem is this: Prove that if p is a prime, then the field Zp is not algebraically closed. I know that using Fermat's little theorem will help but I can't see how it's not closed. Can anybody help please?

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    May 2008
    Posts
    5
    Using Fermat's little theorem we have x^(p-1)-1=0for any x doesn't equal zero,so the polynomial x^(p-1)-2(suppose p>2)doesn't have the zero points inZp

    if p=2,we can find x^2+x+1 hasn't the zero points in Z2.

    So field Zp is not algebraically closed.



    Quote Originally Posted by Zinners View Post
    Hi,

    My problem is this: Prove that if p is a prime, then the field Zp is not algebraically closed. I know that using Fermat's little theorem will help but I can't see how it's not closed. Can anybody help please?

    Thanks
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by Zinners View Post
    Hi,

    My problem is this: Prove that if p is a prime, then the field Zp is not algebraically closed. I know that using Fermat's little theorem will help but I can't see how it's not closed. Can anybody help please?

    Thanks
    In general let F be a finite field with q elements.
    Define f(x) = x^q - x +1, we know that a^{q-1}=1 \implies a^q - a = 0 for all a\in F^{\times}.
    Therefore, f(x) = x^q - x + 1 always has no zero.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 5
    Last Post: June 1st 2011, 01:40 PM
  2. Replies: 1
    Last Post: February 19th 2011, 04:19 AM
  3. Is any algebraically closed field infinite ?
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: January 26th 2011, 12:11 AM
  4. Replies: 6
    Last Post: January 10th 2010, 08:49 AM
  5. Algebraically closed field
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 8th 2008, 03:18 PM

Search Tags


/mathhelpforum @mathhelpforum