Results 1 to 5 of 5
Like Tree2Thanks
  • 1 Post By Plato
  • 1 Post By HallsofIvy

Math Help - Finite field

  1. #1
    Member
    Joined
    Jun 2013
    From
    Israel
    Posts
    158

    Finite field

    So I uploaded a picture that is written a claim that I didnt understand. This is the way that is written in my notebook and I didnt understand why its not a field.
    So from what I have been told that it is not field because some numbers in Zn their inverse. But how is this proof , proving that exactly?
    And I wanted to ask that if this is a contradiction to the uniqueness of 0? I mean in multiplication only when we multiply with zero we get zero but here we didnt multiply with zero is that contradiction to that? thank you
    Attached Thumbnails Attached Thumbnails Finite field-finite-field-claim.png  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,649
    Thanks
    1597
    Awards
    1

    Re: Finite field

    Quote Originally Posted by davidciprut View Post
    So I uploaded a picture that is written a claim that I didnt understand. This is the way that is written in my notebook and I didnt understand why its not a field. So from what I have been told that it is not field because some numbers in Zn their inverse. But how is this proof , proving that exactly? And I wanted to ask that if this is a contradiction to the uniqueness of 0? I mean in multiplication only when we multiply with zero we get zero but here we didnt multiply with zero is that contradiction to that? thank you
    It simply says that if n is not prime then \mathbb{Z}_n is not a field.
    You will have non-zero divisors of zero.

    In \mathbb{Z}_{12} we have 2\cdot 6=0.

    BUT in \mathbb{Z}_{13} and a\ne 0~\&~b\ne 0 we have a\cdot b \ne 0
    Thanks from davidciprut
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jun 2013
    From
    Israel
    Posts
    158

    Re: Finite field

    So the reason is it's a contradiction to the uniqueness of 0?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,693
    Thanks
    1466

    Re: Finite field

    No, it contradicts the fact that every non-zero member has a multiplicative inverse, which itself implies that there cannot be 'zero divisors', non-zero numbers whose product is 0. In particular if 6 had an inverse in Z_{12}, then multiplying both sides of 6*2= 0 would give 6^{-1}(6*2)= (6^{-1}*6)2= 1*2= 2= 0 which is false.
    Thanks from davidciprut
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98

    Re: Finite field

    Def: Zn is a field if ax=b(modn) has a unique solution (modn) for a≠0.

    Theorem: Zn is a field if n =p, p prime.

    It follows that: ab=0(modp), a≠0, -> b=0.

    Def: If ab=0(modn), a≠0, b≠0, a and b are divisors of zero.

    There are no divisors of zero in Zp.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Splitting Field of a Polynomial over a Finite Field
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 1st 2011, 03:45 PM
  2. Finite Field
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: January 15th 2010, 02:25 AM
  3. finite field
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: September 20th 2009, 02:58 AM
  4. Finite field
    Posted in the Advanced Algebra Forum
    Replies: 18
    Last Post: August 28th 2009, 07:26 PM
  5. Finite Field
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: February 9th 2009, 09:08 PM

Search Tags


/mathhelpforum @mathhelpforum