Results 1 to 6 of 6

Math Help - subfield isomorphism

  1. #1
    Member
    Joined
    Nov 2006
    Posts
    142

    subfield isomorphism

    How do I show that if p > 0 is the characteristic of the finite field F (p is prime obviously) that F contains a subfield {k * 1F: k=0,1,...,p-1} that is isomorphic to Zp?

    Note k*1F= 1+1+1...+1 (k times)
    Thank you in advance.

    Also, how can I show that the set {k * 1F: k=0,1,...,p-1} actually is a subfield?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by PvtBillPilgrim View Post
    How do I show that if p > 0 is the characteristic of the finite field F (p is prime obviously) that F contains a subfield {k * 1F: k=0,1,...,p-1} that is isomorphic to Zp?
    This is one of the elegant results about fields.

    Theorem: If R is a ring with unity and with charachteristic n>0, then R contains \mathbb{Z}_n as a subring up to isomorphism.

    Thus, given a field with a charachetristic n>1 then n must be a prime. Because otherwise it will have zero-divisors, which a field does not have. Thus it contains \mathbb{Z}_p as a subfield up to isomorphism.

    Quote Originally Posted by PvtBillPilgrim View Post
    Also, how can I show that the set {k * 1F: k=0,1,...,p-1} actually is a subfield?
    This is a known fact that \mathbb{Z}_p is always a field. Because it is a finite integral domain, and a finite integral domain is always a field.
    Last edited by ThePerfectHacker; January 28th 2007 at 06:07 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Nov 2006
    Posts
    142
    How do you prove that initial part though (the theorem)?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by PvtBillPilgrim View Post
    How do you prove that initial part though (the theorem)?
    The map \phi:\mathbb{Z}\to R which we define as \phi(m)=m\cdot 1 is a homomorphism (excerise, prove it). By the fundamental theorem the kernel is an ideal of \mathbb{Z}. Now all ideals of \mathbb{Z} are of the form k\mathbb{Z} because a subgroup of a cyclic group is cyclic. Thus, if R has charachteristic n>0 then, \ker \phi =n\mathbb{Z}. Again by the fundamental theorem, \phi [\mathbb{Z} ] \simeq \mathbb{Z}/n\mathbb{Z}\simeq \mathbb{Z}_n. Thus, the subring \phi[\mathbb{Z}] confirms the theorem. Q.E.D.

    What is happening to you in this class? I seriously doubt it they will ask you to prove this theorem, is it not in your textbook?
    Last edited by ThePerfectHacker; January 27th 2007 at 09:06 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Nov 2006
    Posts
    142
    Yeah, we have no textbook.

    Thanks for the help.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by PvtBillPilgrim View Post
    Yeah, we have no textbook.
    Why not? Is the professor insane?
    Last edited by ThePerfectHacker; January 28th 2007 at 06:18 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. subfield
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: May 31st 2010, 05:54 PM
  2. Replies: 4
    Last Post: February 14th 2010, 04:05 AM
  3. Subfield of C.
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: February 9th 2010, 09:42 AM
  4. Subfield Q
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 17th 2009, 07:54 PM
  5. subfield
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: December 15th 2008, 08:27 PM

Search Tags


/mathhelpforum @mathhelpforum